Theorem poimirlem4 34039

Description: Lemma for poimir 34068 connecting the admissible faces on the back face of the (𝑀 + 1)-cube to admissible simplices in the 𝑀-cube. (Contributed by Brendan Leahy, 21-Aug-2020.)

Hypotheses

Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem4.1	⊢ (𝜑 → 𝐾 ∈ ℕ)
poimirlem4.2	⊢ (𝜑 → 𝑀 ∈ ℕ0)
poimirlem4.3	⊢ (𝜑 → 𝑀 < 𝑁)

Assertion

Ref

Expression

poimirlem4

⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})

Distinct variable groups:   𝑓,𝑖,𝑗,𝑝,𝑠   𝜑,𝑗   𝑗,𝑀   𝑗,𝑁   𝜑,𝑖,𝑝,𝑠   𝐵,𝑓,𝑖,𝑗,𝑠   𝑓,𝐾,𝑖,𝑗,𝑝,𝑠   𝑓,𝑀,𝑖,𝑝,𝑠   𝑓,𝑁,𝑖,𝑝,𝑠

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)

Proof of Theorem poimirlem4

Dummy variables 𝑘 𝑛 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimir.0	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑁 ∈ ℕ)
3		poimirlem4.1	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℕ)
4	3	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝐾 ∈ ℕ)
5		poimirlem4.2	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
6	5	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑀 ∈ ℕ0)
7		poimirlem4.3	. . . . . . . 8 ⊢ (𝜑 → 𝑀 < 𝑁)
8	7	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑀 < 𝑁)
9		xp1st 7477	. . . . . . . . 9 ⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st ‘𝑡) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)))
10		elmapi 8162	. . . . . . . . 9 ⊢ ((1st ‘𝑡) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) → (1st ‘𝑡):(1...𝑀)⟶(0..^𝐾))
11	9, 10	syl 17	. . . . . . . 8 ⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st ‘𝑡):(1...𝑀)⟶(0..^𝐾))
12	11	adantl 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (1st ‘𝑡):(1...𝑀)⟶(0..^𝐾))
13		xp2nd 7478	. . . . . . . . 9 ⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd ‘𝑡) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})
14		fvex 6459	. . . . . . . . . 10 ⊢ (2nd ‘𝑡) ∈ V
15		f1oeq1 6380	. . . . . . . . . 10 ⊢ (𝑓 = (2nd ‘𝑡) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀)))
16	14, 15	elab 3558	. . . . . . . . 9 ⊢ ((2nd ‘𝑡) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ (2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀))
17	13, 16	sylib 210	. . . . . . . 8 ⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀))
18	17	adantl 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀))
19	2, 4, 6, 8, 12, 18	poimirlem3 34038	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))
20		fvex 6459	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘𝑡) ∈ V
21		snex 5140	. . . . . . . . . . . . . . . 16 ⊢ {⟨(𝑀 + 1), 0⟩} ∈ V
22	20, 21	unex 7233	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∈ V
23		snex 5140	. . . . . . . . . . . . . . . 16 ⊢ {⟨(𝑀 + 1), (𝑀 + 1)⟩} ∈ V
24	14, 23	unex 7233	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V
25	22, 24	op1std 7455	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (1st ‘𝑠) = ((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}))
26	22, 24	op2ndd 7456	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (2nd ‘𝑠) = ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
27	26	imaeq1d 5719	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((2nd ‘𝑠) “ (1...𝑗)) = (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)))
28	27	xpeq1d 5384	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}))
29	26	imaeq1d 5719	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) = (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
30	29	xpeq1d 5384	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
31	28, 30	uneq12d 3991	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})))
32	25, 31	oveq12d 6940	. . . . . . . . . . . . 13 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))))
33	32	uneq1d 3989	. . . . . . . . . . . 12 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
34	33	csbeq1d 3758	. . . . . . . . . . 11 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋((((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
35	34	eqeq2d 2788	. . . . . . . . . 10 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋((((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
36	35	rexbidv 3237	. . . . . . . . 9 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
37	36	ralbidv 3168	. . . . . . . 8 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
38	25	fveq1d 6448	. . . . . . . . 9 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((1st ‘𝑠)‘(𝑀 + 1)) = (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)))
39	38	eqeq1d 2780	. . . . . . . 8 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((1st ‘𝑠)‘(𝑀 + 1)) = 0 ↔ (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0))
40	26	fveq1d 6448	. . . . . . . . 9 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((2nd ‘𝑠)‘(𝑀 + 1)) = (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)))
41	40	eqeq1d 2780	. . . . . . . 8 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1) ↔ (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))
42	37, 39, 41	3anbi123d 1509	. . . . . . 7 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
43	42	elrab 3572	. . . . . 6 ⊢ (⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ (⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
44	19, 43	syl6ibr 244	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}))
45	44	ralrimiva 3148	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}))
46		fveq2 6446	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (1st ‘𝑠) = (1st ‘𝑡))
47		fveq2 6446	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (2nd ‘𝑠) = (2nd ‘𝑡))
48	47	imaeq1d 5719	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd ‘𝑡) “ (1...𝑗)))
49	48	xpeq1d 5384	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = (((2nd ‘𝑡) “ (1...𝑗)) × {1}))
50	47	imaeq1d 5719	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)))
51	50	xpeq1d 5384	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))
52	49, 51	uneq12d 3991	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0})) = ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0})))
53	46, 52	oveq12d 6940	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) = ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))))
54	53	uneq1d 3989	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})))
55	54	csbeq1d 3758	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
56	55	eqeq2d 2788	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
57	56	rexbidv 3237	. . . . . 6 ⊢ (𝑠 = 𝑡 → (∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
58	57	ralbidv 3168	. . . . 5 ⊢ (𝑠 = 𝑡 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
59	58	ralrab 3578	. . . 4 ⊢ (∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ ∀𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}))
60	45, 59	sylibr 226	. . 3 ⊢ (𝜑 → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
61		xp1st 7477	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st ‘𝑘) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))))
62		elmapi 8162	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑘) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) → (1st ‘𝑘):(1...(𝑀 + 1))⟶(0..^𝐾))
63	61, 62	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st ‘𝑘):(1...(𝑀 + 1))⟶(0..^𝐾))
64		fzssp1 12701	. . . . . . . . . . . . . 14 ⊢ (1...𝑀) ⊆ (1...(𝑀 + 1))
65		fssres 6320	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑘):(1...(𝑀 + 1))⟶(0..^𝐾) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((1st ‘𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾))
66	63, 64, 65	sylancl 580	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st ‘𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾))
67		ovex 6954	. . . . . . . . . . . . . 14 ⊢ (0..^𝐾) ∈ V
68		ovex 6954	. . . . . . . . . . . . . 14 ⊢ (1...𝑀) ∈ V
69	67, 68	elmap 8169	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) ↔ ((1st ‘𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾))
70	66, 69	sylibr 226	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st ‘𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)))
71	70	ad2antlr 717	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((1st ‘𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)))
72		xp2nd 7478	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd ‘𝑘) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})
73		fvex 6459	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘𝑘) ∈ V
74		f1oeq1 6380	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (2nd ‘𝑘) → (𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
75	73, 74	elab 3558	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘𝑘) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)))
76	72, 75	sylib 210	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)))
77		f1of1 6390	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd ‘𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1)))
78	76, 77	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd ‘𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1)))
79		f1ores 6405	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1)) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd ‘𝑘) “ (1...𝑀)))
80	78, 64, 79	sylancl 580	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd ‘𝑘) “ (1...𝑀)))
81	80	ad2antlr 717	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd ‘𝑘) “ (1...𝑀)))
82		dff1o3 6397	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ ((2nd ‘𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)) ∧ Fun ◡(2nd ‘𝑘)))
83	82	simprbi 492	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → Fun ◡(2nd ‘𝑘))
84		imadif 6218	. . . . . . . . . . . . . . . . . 18 ⊢ (Fun ◡(2nd ‘𝑘) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd ‘𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd ‘𝑘) “ {(𝑀 + 1)})))
85	76, 83, 84	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd ‘𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd ‘𝑘) “ {(𝑀 + 1)})))
86	85	ad2antlr 717	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd ‘𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd ‘𝑘) “ {(𝑀 + 1)})))
87		f1ofo 6398	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd ‘𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)))
88		foima 6371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)) → ((2nd ‘𝑘) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
89	76, 87, 88	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd ‘𝑘) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
90	89	ad2antlr 717	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
91		f1ofn 6392	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd ‘𝑘) Fn (1...(𝑀 + 1)))
92	76, 91	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd ‘𝑘) Fn (1...(𝑀 + 1)))
93		nn0p1nn 11683	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ)
94	5, 93	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
95		elfz1end 12688	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 + 1) ∈ ℕ ↔ (𝑀 + 1) ∈ (1...(𝑀 + 1)))
96	94, 95	sylib 210	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + 1) ∈ (1...(𝑀 + 1)))
97		fnsnfv 6518	. . . . . . . . . . . . . . . . . . 19 ⊢ (((2nd ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → {((2nd ‘𝑘)‘(𝑀 + 1))} = ((2nd ‘𝑘) “ {(𝑀 + 1)}))
98	92, 96, 97	syl2anr 590	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → {((2nd ‘𝑘)‘(𝑀 + 1))} = ((2nd ‘𝑘) “ {(𝑀 + 1)}))
99		sneq 4408	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {((2nd ‘𝑘)‘(𝑀 + 1))} = {(𝑀 + 1)})
100	98, 99	sylan9req 2835	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ {(𝑀 + 1)}) = {(𝑀 + 1)})
101	90, 100	difeq12d 3952	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd ‘𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd ‘𝑘) “ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}))
102	86, 101	eqtrd 2814	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}))
103		1z 11759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℤ
104		nn0uz 12028	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ0 = (ℤ≥‘0)
105		1m1e0 11447	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 − 1) = 0
106	105	fveq2i 6449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
107	104, 106	eqtr4i 2805	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
108	5, 107	syl6eleq 2869	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(1 − 1)))
109		fzsuc2 12716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ (ℤ≥‘(1 − 1))) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
110	103, 108, 109	sylancr 581	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
111	110	difeq1d 3950	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (((1...𝑀) ∪ {(𝑀 + 1)}) ∖ {(𝑀 + 1)}))
112		difun2 4272	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1...𝑀) ∪ {(𝑀 + 1)}) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)})
113	111, 112	syl6eq 2830	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)}))
114	5	nn0red 11703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑀 ∈ ℝ)
115		ltp1 11215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℝ → 𝑀 < (𝑀 + 1))
116		peano2re 10549	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
117		ltnle 10456	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑀 ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ) → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
118	116, 117	mpdan 677	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℝ → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
119	115, 118	mpbid 224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℝ → ¬ (𝑀 + 1) ≤ 𝑀)
120	114, 119	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
121		elfzle2 12662	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 + 1) ∈ (1...𝑀) → (𝑀 + 1) ≤ 𝑀)
122	120, 121	nsyl 138	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ (𝑀 + 1) ∈ (1...𝑀))
123		difsn 4560	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝑀 + 1) ∈ (1...𝑀) → ((1...𝑀) ∖ {(𝑀 + 1)}) = (1...𝑀))
124	122, 123	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1...𝑀) ∖ {(𝑀 + 1)}) = (1...𝑀))
125	113, 124	eqtrd 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀))
126	125	imaeq2d 5720	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd ‘𝑘) “ (1...𝑀)))
127	126	ad2antrr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd ‘𝑘) “ (1...𝑀)))
128	125	ad2antrr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀))
129	102, 127, 128	3eqtr3d 2822	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ (1...𝑀)) = (1...𝑀))
130		f1oeq3 6382	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑘) “ (1...𝑀)) = (1...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd ‘𝑘) “ (1...𝑀)) ↔ ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀)))
131	129, 130	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd ‘𝑘) “ (1...𝑀)) ↔ ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀)))
132	81, 131	mpbid 224	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀))
133	73	resex 5693	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑘) ↾ (1...𝑀)) ∈ V
134		f1oeq1 6380	. . . . . . . . . . . . 13 ⊢ (𝑓 = ((2nd ‘𝑘) ↾ (1...𝑀)) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀)))
135	133, 134	elab 3558	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝑘) ↾ (1...𝑀)) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ ((2nd ‘𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀))
136	132, 135	sylibr 226	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) ↾ (1...𝑀)) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})
137		opelxpi 5392	. . . . . . . . . . 11 ⊢ ((((1st ‘𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) ∧ ((2nd ‘𝑘) ↾ (1...𝑀)) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
138	71, 136, 137	syl2anc 579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
139	138	3ad2antr3 1198	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
140		3anass 1079	. . . . . . . . . . 11 ⊢ ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))))
141		ancom 454	. . . . . . . . . . 11 ⊢ ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ↔ ((((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
142	140, 141	bitri 267	. . . . . . . . . 10 ⊢ ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ ((((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
143	94	nnzd 11833	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
144		uzid 12007	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑀 + 1) ∈ ℤ → (𝑀 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
145		peano2uz 12047	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘(𝑀 + 1)) → ((𝑀 + 1) + 1) ∈ (ℤ≥‘(𝑀 + 1)))
146	143, 144, 145	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝑀 + 1) + 1) ∈ (ℤ≥‘(𝑀 + 1)))
147	5	nn0zd 11832	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑀 ∈ ℤ)
148	1	nnzd 11833	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑁 ∈ ℤ)
149		zltp1le 11779	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
150		peano2z 11770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ)
151		eluz 12006	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
152	150, 151	sylan 575	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
153	149, 152	bitr4d 274	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))))
154	147, 148, 153	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))))
155	7, 154	mpbid 224	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))
156		fzsplit2 12683	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑀 + 1) + 1) ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
157	146, 155, 156	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
158		fzsn 12700	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑀 + 1) ∈ ℤ → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
159	143, 158	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
160	159	uneq1d 3989	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
161	157, 160	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
162	161	xpeq1d 5384	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}))
163	162	uneq2d 3990	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})))
164		xpundir 5418	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
165		ovex 6954	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 + 1) ∈ V
166		c0ex 10370	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ V
167	165, 166	xpsn 6672	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({(𝑀 + 1)} × {0}) = {⟨(𝑀 + 1), 0⟩}
168	167	uneq1i 3986	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
169	164, 168	eqtri 2802	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
170	169	uneq2i 3987	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
171		unass 3993	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
172	170, 171	eqtr4i 2805	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
173	163, 172	syl6eq 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
174	173	ad3antrrr 720	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
175	165	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ V)
176	166	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ V)
177	110	eqcomd 2784	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
178	177	ad3antrrr 720	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
179		fveq2 6446	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = (𝑀 + 1) → ((1st ‘𝑘)‘𝑛) = ((1st ‘𝑘)‘(𝑀 + 1)))
180		fveq2 6446	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = (𝑀 + 1) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)))
181	179, 180	oveq12d 6940	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = (𝑀 + 1) → (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = (((1st ‘𝑘)‘(𝑀 + 1)) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1))))
182		simplrl 767	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((1st ‘𝑘)‘(𝑀 + 1)) = 0)
183		imain 6219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Fun ◡(2nd ‘𝑘) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
184	76, 83, 183	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
185	184	ad2antlr 717	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
186		elfznn0 12751	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
187	186	nn0red 11703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
188	187	ltp1d 11308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 < (𝑗 + 1))
189		fzdisj 12685	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
190	188, 189	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
191	190	imaeq2d 5720	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd ‘𝑘) “ ∅))
192		ima0 5735	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2nd ‘𝑘) “ ∅) = ∅
193	191, 192	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
194	185, 193	sylan9req 2835	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
195		simplr 759	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))
196	92	ad2antlr 717	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd ‘𝑘) Fn (1...(𝑀 + 1)))
197		nn0p1nn 11683	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ ℕ)
198		nnuz 12029	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ℕ = (ℤ≥‘1)
199	197, 198	syl6eleq 2869	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ (ℤ≥‘1))
200		fzss1 12697	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑗 + 1) ∈ (ℤ≥‘1) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)))
201	186, 199, 200	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)))
202		elfzuz3 12656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ≥‘𝑗))
203		eluzp1p1 12018	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ (ℤ≥‘𝑗) → (𝑀 + 1) ∈ (ℤ≥‘(𝑗 + 1)))
204		eluzfz2 12666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘(𝑗 + 1)) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
205	202, 203, 204	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
206	201, 205	jca 507	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))))
207		fnfvima 6769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((2nd ‘𝑘) Fn (1...(𝑀 + 1)) ∧ ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))) → ((2nd ‘𝑘)‘(𝑀 + 1)) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
208	207	3expb 1110	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((2nd ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))) → ((2nd ‘𝑘)‘(𝑀 + 1)) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
209	196, 206, 208	syl2an 589	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd ‘𝑘)‘(𝑀 + 1)) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
210	195, 209	eqeltrrd 2860	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
211		1ex 10372	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ V
212		fnconstg 6343	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (1 ∈ V → (((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd ‘𝑘) “ (1...𝑗)))
213	211, 212	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd ‘𝑘) “ (1...𝑗))
214		fnconstg 6343	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0 ∈ V → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
215	166, 214	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))
216		fvun2 6530	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd ‘𝑘) “ (1...𝑗)) ∧ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) ∧ ((((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
217	213, 215, 216	mp3an12 1524	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
218	194, 210, 217	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
219	166	fvconst2 6741	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑀 + 1) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) → ((((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
220	210, 219	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
221	218, 220	eqtrd 2814	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = 0)
222	221	adantlrl 710	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = 0)
223	182, 222	oveq12d 6940	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘)‘(𝑀 + 1)) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1))) = (0 + 0))
224		00id 10551	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 + 0) = 0
225	223, 224	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘)‘(𝑀 + 1)) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1))) = 0)
226	181, 225	sylan9eqr 2836	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = 0)
227	175, 176, 178, 226	fmptapd 6704	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
228	227	uneq1d 3989	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
229	174, 228	eqtrd 2814	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
230		elmapfn 8163	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1st ‘𝑘) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) → (1st ‘𝑘) Fn (1...(𝑀 + 1)))
231	61, 230	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st ‘𝑘) Fn (1...(𝑀 + 1)))
232		fnssres 6250	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((1st ‘𝑘) ↾ (1...𝑀)) Fn (1...𝑀))
233	231, 64, 232	sylancl 580	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st ‘𝑘) ↾ (1...𝑀)) Fn (1...𝑀))
234	233	ad3antlr 721	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1st ‘𝑘) ↾ (1...𝑀)) Fn (1...𝑀))
235		simplr 759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}))
236		fnconstg 6343	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0 ∈ V → (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)))
237	166, 236	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))
238	213, 237	pm3.2i 464	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd ‘𝑘) “ (1...𝑗)) ∧ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)))
239		imain 6219	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (Fun ◡(2nd ‘𝑘) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))))
240	76, 83, 239	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))))
241		fzdisj 12685	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
242	188, 241	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
243	242	imaeq2d 5720	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((2nd ‘𝑘) “ ∅))
244	243, 192	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ∅)
245	240, 244	sylan9req 2835	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))) = ∅)
246		fnun 6243	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd ‘𝑘) “ (1...𝑗)) ∧ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))) ∧ (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))) = ∅) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))))
247	238, 245, 246	sylancr 581	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))))
248	235, 247	sylan 575	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))))
249	101	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd ‘𝑘) “ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}))
250	85	ad3antlr 721	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd ‘𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd ‘𝑘) “ {(𝑀 + 1)})))
251	186, 197	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℕ)
252	251, 198	syl6eleq 2869	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ (ℤ≥‘1))
253		fzsplit2 12683	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑗 + 1) ∈ (ℤ≥‘1) ∧ 𝑀 ∈ (ℤ≥‘𝑗)) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
254	252, 202, 253	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑀) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
255	128, 254	sylan9eq 2834	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
256	255	imaeq2d 5720	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
257	250, 256	eqtr3d 2816	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd ‘𝑘) “ {(𝑀 + 1)})) = ((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
258	125	ad3antrrr 720	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀))
259	249, 257, 258	3eqtr3rd 2823	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) = ((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
260		imaundi 5799	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)))
261	259, 260	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) = (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))))
262	261	fneq2d 6227	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ↔ ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)))))
263	248, 262	mpbird 249	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀))
264		fzss2 12698	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑀 ∈ (ℤ≥‘𝑗) → (1...𝑗) ⊆ (1...𝑀))
265		resima2 5681	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1...𝑗) ⊆ (1...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) = ((2nd ‘𝑘) “ (1...𝑗)))
266	202, 264, 265	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) = ((2nd ‘𝑘) “ (1...𝑗)))
267	266	xpeq1d 5384	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) = (((2nd ‘𝑘) “ (1...𝑗)) × {1}))
268	186, 199	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ (ℤ≥‘1))
269		fzss1 12697	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑗 + 1) ∈ (ℤ≥‘1) → ((𝑗 + 1)...𝑀) ⊆ (1...𝑀))
270		resima2 5681	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑗 + 1)...𝑀) ⊆ (1...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)))
271	268, 269, 270	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)))
272	271	xpeq1d 5384	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))
273	267, 272	uneq12d 3991	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ (0...𝑀) → (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) = ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})))
274	273	adantl 475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) = ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})))
275	274	fneq1d 6226	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ↔ ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀)))
276	263, 275	mpbird 249	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀))
277		fzfid 13091	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) ∈ Fin)
278		inidm 4043	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1...𝑀) ∩ (1...𝑀)) = (1...𝑀)
279		fvres 6465	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑀) → (((1st ‘𝑘) ↾ (1...𝑀))‘𝑛) = ((1st ‘𝑘)‘𝑛))
280	279	adantl 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((1st ‘𝑘) ↾ (1...𝑀))‘𝑛) = ((1st ‘𝑘)‘𝑛))
281		disjsn 4478	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ↔ ¬ (𝑀 + 1) ∈ (1...𝑀))
282	122, 281	sylibr 226	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)
283	282	ad3antrrr 720	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)
284	263, 283	jca 507	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅))
285		fnconstg 6343	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 ∈ V → ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)})
286	166, 285	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)}
287		fvun1 6529	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
288	286, 287	mp3an2 1522	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
289	288	anassrs 461	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
290	284, 289	sylan 575	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
291	251	nnzd 11833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℤ)
292	186	nn0cnd 11704	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ)
293		pncan1 10799	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑗 ∈ ℂ → ((𝑗 + 1) − 1) = 𝑗)
294	292, 293	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1) − 1) = 𝑗)
295	294	fveq2d 6450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑗 ∈ (0...𝑀) → (ℤ≥‘((𝑗 + 1) − 1)) = (ℤ≥‘𝑗))
296	202, 295	eleqtrrd 2862	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ≥‘((𝑗 + 1) − 1)))
297		fzsuc2 12716	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑗 + 1) ∈ ℤ ∧ 𝑀 ∈ (ℤ≥‘((𝑗 + 1) − 1))) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)}))
298	291, 296, 297	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)}))
299	298	imaeq2d 5720	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd ‘𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})))
300		imaundi 5799	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((2nd ‘𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})) = (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)}))
301	299, 300	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})))
302	301	xpeq1d 5384	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})) × {0}))
303		xpundir 5418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})) × {0}) = ((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0}))
304	302, 303	syl6eq 2830	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0})))
305	304	uneq2d 3990	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0}))))
306		unass 3993	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0})) = ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0})))
307	305, 306	syl6eqr 2832	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0})))
308	307	adantl 475	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0})))
309	98	xpeq1d 5384	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ({((2nd ‘𝑘)‘(𝑀 + 1))} × {0}) = (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0}))
310	309	uneq2d 3990	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd ‘𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0})))
311	310	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd ‘𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0})))
312	308, 311	eqtr4d 2817	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd ‘𝑘)‘(𝑀 + 1))} × {0})))
313	99	xpeq1d 5384	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → ({((2nd ‘𝑘)‘(𝑀 + 1))} × {0}) = ({(𝑀 + 1)} × {0}))
314	313	uneq2d 3990	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd ‘𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0})))
315	312, 314	sylan9eq 2834	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0})))
316	315	an32s 642	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0})))
317	316	fveq1d 6448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛))
318	317	adantr 474	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛))
319	273	fveq1d 6448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ (0...𝑀) → ((((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
320	319	ad2antlr 717	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
321	290, 318, 320	3eqtr4rd 2825	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))
322	234, 276, 277, 277, 278, 280, 321	offval 7181	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) = (𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
323	322	uneq1d 3989	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})))
324	323	adantlrl 710	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})))
325		simplr 759	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}))
326	231	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (1st ‘𝑘) Fn (1...(𝑀 + 1)))
327	213, 215	pm3.2i 464	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd ‘𝑘) “ (1...𝑗)) ∧ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
328	184, 193	sylan9req 2835	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
329		fnun 6243	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd ‘𝑘) “ (1...𝑗)) ∧ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) ∧ (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
330	327, 328, 329	sylancr 581	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
331		peano2uz 12047	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ (ℤ≥‘𝑗) → (𝑀 + 1) ∈ (ℤ≥‘𝑗))
332	202, 331	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑗))
333		fzsplit2 12683	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑗 + 1) ∈ (ℤ≥‘1) ∧ (𝑀 + 1) ∈ (ℤ≥‘𝑗)) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
334	268, 332, 333	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑀) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
335	334	imaeq2d 5720	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ (1...(𝑀 + 1))) = ((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))))
336		imaundi 5799	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
337	335, 336	syl6req 2831	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd ‘𝑘) “ (1...(𝑀 + 1))))
338	337, 89	sylan9eqr 2836	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = (1...(𝑀 + 1)))
339	338	fneq2d 6227	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) ↔ ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1))))
340	330, 339	mpbid 224	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1)))
341		fzfid 13091	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) ∈ Fin)
342		inidm 4043	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1...(𝑀 + 1)) ∩ (1...(𝑀 + 1))) = (1...(𝑀 + 1))
343		eqidd 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → ((1st ‘𝑘)‘𝑛) = ((1st ‘𝑘)‘𝑛))
344		eqidd 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))
345	326, 340, 341, 341, 342, 343, 344	offval 7181	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
346	345	uneq1d 3989	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
347	325, 346	sylan 575	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st ‘𝑘)‘𝑛) + (((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
348	229, 324, 347	3eqtr4rd 2825	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})))
349	348	csbeq1d 3758	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
350	349	eqeq2d 2788	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
351	350	rexbidva 3234	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
352	351	ralbidv 3168	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
353	352	biimpd 221	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
354	353	impr 448	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)) → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
355	142, 354	sylan2b 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
356		1st2nd2 7484	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → 𝑘 = ⟨(1st ‘𝑘), (2nd ‘𝑘)⟩)
357	356	ad2antlr 717	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = ⟨(1st ‘𝑘), (2nd ‘𝑘)⟩)
358		fnsnsplit 6721	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → (1st ‘𝑘) = (((1st ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))⟩}))
359	231, 96, 358	syl2anr 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (1st ‘𝑘) = (((1st ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))⟩}))
360	359	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0) → (1st ‘𝑘) = (((1st ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))⟩}))
361	125	reseq2d 5642	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1st ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1st ‘𝑘) ↾ (1...𝑀)))
362	361	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((1st ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1st ‘𝑘) ↾ (1...𝑀)))
363		opeq2 4637	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 → ⟨(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))⟩ = ⟨(𝑀 + 1), 0⟩)
364	363	sneqd 4410	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 → {⟨(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), 0⟩})
365		uneq12 3985	. . . . . . . . . . . . . . 15 ⊢ ((((1st ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1st ‘𝑘) ↾ (1...𝑀)) ∧ {⟨(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), 0⟩}) → (((1st ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))⟩}) = (((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
366	362, 364, 365	syl2an 589	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0) → (((1st ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))⟩}) = (((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
367	360, 366	eqtrd 2814	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0) → (1st ‘𝑘) = (((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
368	367	adantrr 707	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (1st ‘𝑘) = (((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
369		fnsnsplit 6721	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → (2nd ‘𝑘) = (((2nd ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))⟩}))
370	92, 96, 369	syl2anr 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (2nd ‘𝑘) = (((2nd ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))⟩}))
371	370	adantr 474	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd ‘𝑘) = (((2nd ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))⟩}))
372	125	reseq2d 5642	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd ‘𝑘) ↾ (1...𝑀)))
373	372	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((2nd ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd ‘𝑘) ↾ (1...𝑀)))
374		opeq2 4637	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → ⟨(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))⟩ = ⟨(𝑀 + 1), (𝑀 + 1)⟩)
375	374	sneqd 4410	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {⟨(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), (𝑀 + 1)⟩})
376		uneq12 3985	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd ‘𝑘) ↾ (1...𝑀)) ∧ {⟨(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → (((2nd ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))⟩}) = (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
377	373, 375, 376	syl2an 589	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))⟩}) = (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
378	371, 377	eqtrd 2814	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd ‘𝑘) = (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
379	378	adantrl 706	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (2nd ‘𝑘) = (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
380	368, 379	opeq12d 4644	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ⟨(1st ‘𝑘), (2nd ‘𝑘)⟩ = ⟨(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
381	357, 380	eqtrd 2814	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = ⟨(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
382	381	3adantr1 1171	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = ⟨(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
383		fvex 6459	. . . . . . . . . . . . . . . . . . 19 ⊢ (1st ‘𝑘) ∈ V
384	383	resex 5693	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑘) ↾ (1...𝑀)) ∈ V
385	384, 133	op1std 7455	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (1st ‘𝑡) = ((1st ‘𝑘) ↾ (1...𝑀)))
386	384, 133	op2ndd 7456	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (2nd ‘𝑡) = ((2nd ‘𝑘) ↾ (1...𝑀)))
387	386	imaeq1d 5719	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ((2nd ‘𝑡) “ (1...𝑗)) = (((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)))
388	387	xpeq1d 5384	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (((2nd ‘𝑡) “ (1...𝑗)) × {1}) = ((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}))
389	386	imaeq1d 5719	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) = (((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)))
390	389	xpeq1d 5384	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}) = ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))
391	388, 390	uneq12d 3991	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0})) = (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})))
392	385, 391	oveq12d 6940	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) = (((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))))
393	392	uneq1d 3989	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})))
394	393	csbeq1d 3758	. . . . . . . . . . . . . 14 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
395	394	eqeq2d 2788	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
396	395	rexbidv 3237	. . . . . . . . . . . 12 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
397	396	ralbidv 3168	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
398	385	uneq1d 3989	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = (((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
399	386	uneq1d 3989	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
400	398, 399	opeq12d 4644	. . . . . . . . . . . 12 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
401	400	eqeq2d 2788	. . . . . . . . . . 11 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ 𝑘 = ⟨(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
402	397, 401	anbi12d 624	. . . . . . . . . 10 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = ⟨(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)))
403	402	rspcev 3511	. . . . . . . . 9 ⊢ ((⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((((1st ‘𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = ⟨(((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)) → ∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
404	139, 355, 382, 403	syl12anc 827	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
405	404	ex 403	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)))
406		elrabi 3567	. . . . . . . . . . 11 ⊢ (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} → 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
407		elrabi 3567	. . . . . . . . . . 11 ⊢ (𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} → 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
408	406, 407	anim12i 606	. . . . . . . . . 10 ⊢ ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∧ 𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})))
409		eqtr2 2800	. . . . . . . . . . . 12 ⊢ ((𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
410	22, 24	opth 5176	. . . . . . . . . . . . 13 ⊢ (⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
411		difeq1 3944	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) → (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}) = (((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}))
412		difun2 4272	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩})
413		difun2 4272	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩})
414	411, 412, 413	3eqtr3g 2837	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) → ((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}))
415		difeq1 3944	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
416		difun2 4272	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})
417		difun2 4272	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})
418	415, 416, 417	3eqtr3g 2837	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → ((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
419	414, 418	anim12i 606	. . . . . . . . . . . . 13 ⊢ ((((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})) → (((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
420	410, 419	sylbi 209	. . . . . . . . . . . 12 ⊢ (⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
421	409, 420	syl 17	. . . . . . . . . . 11 ⊢ ((𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → (((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
422		elmapfn 8163	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑡) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) → (1st ‘𝑡) Fn (1...𝑀))
423		fnop 6240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1st ‘𝑡) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑡)) → (𝑀 + 1) ∈ (1...𝑀))
424	423	ex 403	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑡) Fn (1...𝑀) → (⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
425	9, 422, 424	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
426	425, 122	nsyli 157	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑡)))
427	426	impcom 398	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑡))
428		difsn 4560	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑡) → ((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st ‘𝑡))
429	427, 428	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st ‘𝑡))
430		xp1st 7477	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st ‘𝑛) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)))
431		elmapfn 8163	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑛) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) → (1st ‘𝑛) Fn (1...𝑀))
432		fnop 6240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1st ‘𝑛) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑛)) → (𝑀 + 1) ∈ (1...𝑀))
433	432	ex 403	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑛) Fn (1...𝑀) → (⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
434	430, 431, 433	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
435	434, 122	nsyli 157	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑛)))
436	435	impcom 398	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑛))
437		difsn 4560	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑛) → ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st ‘𝑛))
438	436, 437	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st ‘𝑛))
439	429, 438	eqeqan12d 2794	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) ∧ (𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ↔ (1st ‘𝑡) = (1st ‘𝑛)))
440	439	anandis 668	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ↔ (1st ‘𝑡) = (1st ‘𝑛)))
441		f1ofn 6392	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd ‘𝑡) Fn (1...𝑀))
442		fnop 6240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘𝑡) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑡)) → (𝑀 + 1) ∈ (1...𝑀))
443	442	ex 403	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑡) Fn (1...𝑀) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
444	17, 441, 443	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
445	444, 122	nsyli 157	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑡)))
446	445	impcom 398	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑡))
447		difsn 4560	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑡) → ((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd ‘𝑡))
448	446, 447	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd ‘𝑡))
449		xp2nd 7478	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd ‘𝑛) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})
450		fvex 6459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2nd ‘𝑛) ∈ V
451		f1oeq1 6380	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = (2nd ‘𝑛) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀)))
452	450, 451	elab 3558	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘𝑛) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ (2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀))
453	449, 452	sylib 210	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀))
454		f1ofn 6392	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd ‘𝑛) Fn (1...𝑀))
455		fnop 6240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘𝑛) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑛)) → (𝑀 + 1) ∈ (1...𝑀))
456	455	ex 403	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑛) Fn (1...𝑀) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
457	453, 454, 456	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
458	457, 122	nsyli 157	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑛)))
459	458	impcom 398	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑛))
460		difsn 4560	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑛) → ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd ‘𝑛))
461	459, 460	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd ‘𝑛))
462	448, 461	eqeqan12d 2794	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) ∧ (𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ↔ (2nd ‘𝑡) = (2nd ‘𝑛)))
463	462	anandis 668	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ↔ (2nd ‘𝑡) = (2nd ‘𝑛)))
464	440, 463	anbi12d 624	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})) ↔ ((1st ‘𝑡) = (1st ‘𝑛) ∧ (2nd ‘𝑡) = (2nd ‘𝑛))))
465		xpopth 7486	. . . . . . . . . . . . 13 ⊢ ((𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (((1st ‘𝑡) = (1st ‘𝑛) ∧ (2nd ‘𝑡) = (2nd ‘𝑛)) ↔ 𝑡 = 𝑛))
466	465	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st ‘𝑡) = (1st ‘𝑛) ∧ (2nd ‘𝑡) = (2nd ‘𝑛)) ↔ 𝑡 = 𝑛))
467	464, 466	bitrd 271	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})) ↔ 𝑡 = 𝑛))
468	421, 467	syl5ib 236	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
469	408, 468	sylan2 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∧ 𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) → ((𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
470	469	ralrimivva 3153	. . . . . . . 8 ⊢ (𝜑 → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
471	470	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
472	405, 471	jctird 522	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))))
473		fveq2 6446	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑛 → (1st ‘𝑡) = (1st ‘𝑛))
474	473	uneq1d 3989	. . . . . . . . . 10 ⊢ (𝑡 = 𝑛 → ((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}))
475		fveq2 6446	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑛 → (2nd ‘𝑡) = (2nd ‘𝑛))
476	475	uneq1d 3989	. . . . . . . . . 10 ⊢ (𝑡 = 𝑛 → ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
477	474, 476	opeq12d 4644	. . . . . . . . 9 ⊢ (𝑡 = 𝑛 → ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
478	477	eqeq2d 2788	. . . . . . . 8 ⊢ (𝑡 = 𝑛 → (𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
479	478	reu4 3612	. . . . . . 7 ⊢ (∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ (∃𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)))
480	58	rexrab 3580	. . . . . . . 8 ⊢ (∃𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ ∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
481	480	anbi1i 617	. . . . . . 7 ⊢ ((∃𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)) ↔ (∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)))
482	479, 481	bitri 267	. . . . . 6 ⊢ (∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ (∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑡) ∘𝑓 + ((((2nd ‘𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ 𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ((𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)))
483	472, 482	syl6ibr 244	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
484	483	ralrimiva 3148	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
485		fveq2 6446	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑘 → (1st ‘𝑠) = (1st ‘𝑘))
486		fveq2 6446	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑘 → (2nd ‘𝑠) = (2nd ‘𝑘))
487	486	imaeq1d 5719	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd ‘𝑘) “ (1...𝑗)))
488	487	xpeq1d 5384	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = (((2nd ‘𝑘) “ (1...𝑗)) × {1}))
489	486	imaeq1d 5719	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
490	489	xpeq1d 5384	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
491	488, 490	uneq12d 3991	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑘 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})))
492	485, 491	oveq12d 6940	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑘 → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = ((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))))
493	492	uneq1d 3989	. . . . . . . . . 10 ⊢ (𝑠 = 𝑘 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = (((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
494	493	csbeq1d 3758	. . . . . . . . 9 ⊢ (𝑠 = 𝑘 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
495	494	eqeq2d 2788	. . . . . . . 8 ⊢ (𝑠 = 𝑘 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
496	495	rexbidv 3237	. . . . . . 7 ⊢ (𝑠 = 𝑘 → (∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
497	496	ralbidv 3168	. . . . . 6 ⊢ (𝑠 = 𝑘 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
498	485	fveq1d 6448	. . . . . . 7 ⊢ (𝑠 = 𝑘 → ((1st ‘𝑠)‘(𝑀 + 1)) = ((1st ‘𝑘)‘(𝑀 + 1)))
499	498	eqeq1d 2780	. . . . . 6 ⊢ (𝑠 = 𝑘 → (((1st ‘𝑠)‘(𝑀 + 1)) = 0 ↔ ((1st ‘𝑘)‘(𝑀 + 1)) = 0))
500	486	fveq1d 6448	. . . . . . 7 ⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠)‘(𝑀 + 1)) = ((2nd ‘𝑘)‘(𝑀 + 1)))
501	500	eqeq1d 2780	. . . . . 6 ⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1) ↔ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)))
502	497, 499, 501	3anbi123d 1509	. . . . 5 ⊢ (𝑠 = 𝑘 → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1))))
503	502	ralrab 3578	. . . 4 ⊢ (∀𝑘 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ ∀𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑘) ∘𝑓 + ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
504	484, 503	sylibr 226	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
505		eqid 2778	. . . 4 ⊢ (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ↦ ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) = (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ↦ ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
506	505	f1ompt 6645	. . 3 ⊢ ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ↦ ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩):{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ (∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ∧ ∀𝑘 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}𝑘 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
507	60, 504, 506	sylanbrc 578	. 2 ⊢ (𝜑 → (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ↦ ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩):{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
508		ovex 6954	. . . . 5 ⊢ ((0..^𝐾) ↑𝑚 (1...𝑀)) ∈ V
509		ovex 6954	. . . . . 6 ⊢ ((1...𝑀) ↑𝑚 (1...𝑀)) ∈ V
510		f1of 6391	. . . . . . . 8 ⊢ (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑓:(1...𝑀)⟶(1...𝑀))
511	510	ss2abi 3895	. . . . . . 7 ⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ {𝑓 ∣ 𝑓:(1...𝑀)⟶(1...𝑀)}
512	68, 68	mapval 8152	. . . . . . 7 ⊢ ((1...𝑀) ↑𝑚 (1...𝑀)) = {𝑓 ∣ 𝑓:(1...𝑀)⟶(1...𝑀)}
513	511, 512	sseqtr4i 3857	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ ((1...𝑀) ↑𝑚 (1...𝑀))
514	509, 513	ssexi 5040	. . . . 5 ⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ∈ V
515	508, 514	xpex 7240	. . . 4 ⊢ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∈ V
516	515	rabex 5049	. . 3 ⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ V
517	516	f1oen 8262	. 2 ⊢ ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ↦ ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩):{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
518	507, 517	syl 17	1 ⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  {cab 2763  ∀wral 3090  ∃wrex 3091  ∃!wreu 3092  {crab 3094  Vcvv 3398  ⦋csb 3751   ∖ cdif 3789   ∪ cun 3790   ∩ cin 3791   ⊆ wss 3792  ∅c0 4141  {csn 4398  ⟨cop 4404   class class class wbr 4886   ↦ cmpt 4965   × cxp 5353  ^◡ccnv 5354   ↾ cres 5357   “ cima 5358  Fun wfun 6129   Fn wfn 6130  ⟶wf 6131  –1-1→wf1 6132  –onto→wfo 6133  –1-1-onto→wf1o 6134  ‘cfv 6135  (class class class)co 6922   ∘_𝑓 cof 7172  1^st c1st 7443  2^nd c2nd 7444   ↑_𝑚 cmap 8140   ≈ cen 8238  Fincfn 8241  ℂcc 10270  ℝcr 10271  0cc0 10272  1c1 10273   + caddc 10275   < clt 10411   ≤ cle 10412   − cmin 10606  ℕcn 11374  ℕ₀cn0 11642  ℤcz 11728  ℤ_≥cuz 11992  ...cfz 12643  ..^cfzo 12784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785

This theorem is referenced by:  poimirlem28  34063