Theorem poimirlem4 35708

Description: Lemma for poimir 35737 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 480	. . . . . . 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 480	. . . . . . 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 480	. . . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑀 < 𝑁)
9		xp1st 7836	. . . . . . . . 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 8595	. . . . . . . . 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 481	. . . . . . 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 7837	. . . . . . . . 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 6769	. . . . . . . . . 10 ⊢ (2nd ‘𝑡) ∈ V
15		f1oeq1 6688	. . . . . . . . . 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 3602	. . . . . . . . 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 217	. . . . . . . 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 481	. . . . . . 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 35707	. . . . . 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 6769	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘𝑡) ∈ V
21		snex 5349	. . . . . . . . . . . . . . . 16 ⊢ {⟨(𝑀 + 1), 0⟩} ∈ V
22	20, 21	unex 7574	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∈ V
23		snex 5349	. . . . . . . . . . . . . . . 16 ⊢ {⟨(𝑀 + 1), (𝑀 + 1)⟩} ∈ V
24	14, 23	unex 7574	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V
25	22, 24	op1std 7814	. . . . . . . . . . . . . 14 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (1st ‘𝑠) = ((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}))
26	22, 24	op2ndd 7815	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = ⟨((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (2nd ‘𝑠) = ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
27	26	imaeq1d 5957	. . . . . . . . . . . . . . . 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 5609	. . . . . . . . . . . . . . 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 5957	. . . . . . . . . . . . . . . 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 5609	. . . . . . . . . . . . . . 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 4094	. . . . . . . . . . . . . 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 7273	. . . . . . . . . . . . 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 4092	. . . . . . . . . . . 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 3832	. . . . . . . . . . 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 2749	. . . . . . . . . 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 3225	. . . . . . . . 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 3120	. . . . . . . 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 6758	. . . . . . . . 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 2740	. . . . . . . 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 6758	. . . . . . . . 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 2740	. . . . . . . 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 1434	. . . . . . 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 3617	. . . . . 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 251	. . . . 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 3107	. . . 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 6756	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (1st ‘𝑠) = (1st ‘𝑡))
47		fveq2 6756	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑡 → (2nd ‘𝑠) = (2nd ‘𝑡))
48	47	imaeq1d 5957	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd ‘𝑡) “ (1...𝑗)))
49	48	xpeq1d 5609	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = (((2nd ‘𝑡) “ (1...𝑗)) × {1}))
50	47	imaeq1d 5957	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑡 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)))
51	50	xpeq1d 5609	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))
52	49, 51	uneq12d 4094	. . . . . . . . . . 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 7273	. . . . . . . . . 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 4092	. . . . . . . . 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 3832	. . . . . . . 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 2749	. . . . . . 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 3225	. . . . . 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 3120	. . . . 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 3623	. . . 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 233	. . 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 7836	. . . . . . . . . . . . . . 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 8595	. . . . . . . . . . . . . . 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 13228	. . . . . . . . . . . . . 14 ⊢ (1...𝑀) ⊆ (1...(𝑀 + 1))
65		fssres 6624	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 7288	. . . . . . . . . . . . . 14 ⊢ (0..^𝐾) ∈ V
68		ovex 7288	. . . . . . . . . . . . . 14 ⊢ (1...𝑀) ∈ V
69	67, 68	elmap 8617	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑m (1...𝑀)) ↔ ((1st ‘𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾))
70	66, 69	sylibr 233	. . . . . . . . . . . 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 723	. . . . . . . . . . 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 7837	. . . . . . . . . . . . . . . . 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 6769	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘𝑘) ∈ V
74		f1oeq1 6688	. . . . . . . . . . . . . . . . . 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 3602	. . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . 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 6699	. . . . . . . . . . . . . . . 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 6714	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 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 723	. . . . . . . . . . . . 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 6706	. . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → Fun ◡(2nd ‘𝑘))
84		imadif 6502	. . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . 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 6707	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd ‘𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)))
88		foima 6677	. . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . 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 6701	. . . . . . . . . . . . . . . . . . . 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 12202	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ)
94	5, 93	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
95		elfz1end 13215	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 + 1) ∈ ℕ ↔ (𝑀 + 1) ∈ (1...(𝑀 + 1)))
96	94, 95	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + 1) ∈ (1...(𝑀 + 1)))
97		fnsnfv 6829	. . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . 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 4568	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {((2nd ‘𝑘)‘(𝑀 + 1))} = {(𝑀 + 1)})
100	98, 99	sylan9req 2800	. . . . . . . . . . . . . . . . 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 4054	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 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 12280	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℤ
104		nn0uz 12549	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℕ0 = (ℤ≥‘0)
105		1m1e0 11975	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 − 1) = 0
106	105	fveq2i 6759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
107	104, 106	eqtr4i 2769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
108	5, 107	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(1 − 1)))
109		fzsuc2 13243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ (ℤ≥‘(1 − 1))) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
110	103, 108, 109	sylancr 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
111	110	difeq1d 4052	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (((1...𝑀) ∪ {(𝑀 + 1)}) ∖ {(𝑀 + 1)}))
112		difun2 4411	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1...𝑀) ∪ {(𝑀 + 1)}) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)})
113	111, 112	eqtrdi 2795	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)}))
114	5	nn0red 12224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑀 ∈ ℝ)
115		ltp1 11745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℝ → 𝑀 < (𝑀 + 1))
116		peano2re 11078	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
117		ltnle 10985	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑀 ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ) → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
118	116, 117	mpdan 683	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℝ → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
119	115, 118	mpbid 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℝ → ¬ (𝑀 + 1) ≤ 𝑀)
120	114, 119	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
121		elfzle2 13189	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 + 1) ∈ (1...𝑀) → (𝑀 + 1) ≤ 𝑀)
122	120, 121	nsyl 140	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ (𝑀 + 1) ∈ (1...𝑀))
123		difsn 4728	. . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀))
126	125	imaeq2d 5958	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd ‘𝑘) “ (1...𝑀)))
127	126	ad2antrr 722	. . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . 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	129	f1oeq3d 6697	. . . . . . . . . . . . 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)))
131	81, 130	mpbid 231	. . . . . . . . . . . 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))
132	73	resex 5928	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑘) ↾ (1...𝑀)) ∈ V
133		f1oeq1 6688	. . . . . . . . . . . . 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)))
134	132, 133	elab 3602	. . . . . . . . . . . 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))
135	131, 134	sylibr 233	. . . . . . . . . . 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)})
136	71, 135	opelxpd 5618	. . . . . . . . . 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)}))
137	136	3ad2antr3 1188	. . . . . . . . 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)}))
138		3anass 1093	. . . . . . . . . . 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))))
139	138	biancomi 462	. . . . . . . . . 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))
140	94	nnzd 12354	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
141		uzid 12526	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑀 + 1) ∈ ℤ → (𝑀 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
142		peano2uz 12570	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘(𝑀 + 1)) → ((𝑀 + 1) + 1) ∈ (ℤ≥‘(𝑀 + 1)))
143	140, 141, 142	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝑀 + 1) + 1) ∈ (ℤ≥‘(𝑀 + 1)))
144	5	nn0zd 12353	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑀 ∈ ℤ)
145	1	nnzd 12354	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝑁 ∈ ℤ)
146		zltp1le 12300	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
147		peano2z 12291	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ)
148		eluz 12525	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
149	147, 148	sylan 579	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
150	146, 149	bitr4d 281	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))))
151	144, 145, 150	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))))
152	7, 151	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))
153		fzsplit2 13210	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑀 + 1) + 1) ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
154	143, 152, 153	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
155		fzsn 13227	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑀 + 1) ∈ ℤ → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
156	140, 155	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
157	156	uneq1d 4092	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
158	154, 157	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
159	158	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}))
160	159	uneq2d 4093	. . . . . . . . . . . . . . . . . . . 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})))
161		xpundir 5647	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
162		ovex 7288	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 + 1) ∈ V
163		c0ex 10900	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ V
164	162, 163	xpsn 6995	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({(𝑀 + 1)} × {0}) = {⟨(𝑀 + 1), 0⟩}
165	164	uneq1i 4089	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
166	161, 165	eqtri 2766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
167	166	uneq2i 4090	. . . . . . . . . . . . . . . . . . . . 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})))
168		unass 4096	. . . . . . . . . . . . . . . . . . . . 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})))
169	167, 168	eqtr4i 2769	. . . . . . . . . . . . . . . . . . . 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}))
170	160, 169	eqtrdi 2795	. . . . . . . . . . . . . . . . . . 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})))
171	170	ad3antrrr 726	. . . . . . . . . . . . . . . . . 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})))
172	162	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)
173	163	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)
174	110	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
175	174	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . 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)))
176		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = (𝑀 + 1) → ((1st ‘𝑘)‘𝑛) = ((1st ‘𝑘)‘(𝑀 + 1)))
177		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . 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)))
178	176, 177	oveq12d 7273	. . . . . . . . . . . . . . . . . . . . 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))))
179		simplrl 773	. . . . . . . . . . . . . . . . . . . . . . 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)
180		imain 6503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Fun ◡(2nd ‘𝑘) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
181	76, 83, 180	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)))))
182	181	ad2antlr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
183		elfznn0 13278	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
184	183	nn0red 12224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
185	184	ltp1d 11835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 < (𝑗 + 1))
186		fzdisj 13212	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
187	185, 186	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
188	187	imaeq2d 5958	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd ‘𝑘) “ ∅))
189		ima0 5974	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((2nd ‘𝑘) “ ∅) = ∅
190	188, 189	eqtrdi 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
191	182, 190	sylan9req 2800	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))) = ∅)
192		simplr 765	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))
193	92	ad2antlr 723	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
194		nn0p1nn 12202	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ ℕ)
195		nnuz 12550	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ℕ = (ℤ≥‘1)
196	194, 195	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ (ℤ≥‘1))
197		fzss1 13224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑗 + 1) ∈ (ℤ≥‘1) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)))
198	183, 196, 197	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)))
199		elfzuz3 13182	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ≥‘𝑗))
200		eluzp1p1 12539	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 ∈ (ℤ≥‘𝑗) → (𝑀 + 1) ∈ (ℤ≥‘(𝑗 + 1)))
201		eluzfz2 13193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘(𝑗 + 1)) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
202	199, 200, 201	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
203	198, 202	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))))
204		fnfvima 7091	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
205	204	3expb 1118	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
206	193, 203, 205	syl2an 595	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
207	192, 206	eqeltrrd 2840	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
208		1ex 10902	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ V
209		fnconstg 6646	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (1 ∈ V → (((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd ‘𝑘) “ (1...𝑗)))
210	208, 209	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd ‘𝑘) “ (1...𝑗))
211		fnconstg 6646	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0 ∈ V → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
212	163, 211	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))
213		fvun2 6842	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
214	210, 212, 213	mp3an12 1449	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))
215	191, 207, 214	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . 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)))
216	163	fvconst2 7061	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑀 + 1) ∈ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) → ((((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
217	207, 216	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)
218	215, 217	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . 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)
219	218	adantlrl 716	. . . . . . . . . . . . . . . . . . . . . . 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)
220	179, 219	oveq12d 7273	. . . . . . . . . . . . . . . . . . . . . 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))
221		00id 11080	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 + 0) = 0
222	220, 221	eqtrdi 2795	. . . . . . . . . . . . . . . . . . . . 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)
223	178, 222	sylan9eqr 2801	. . . . . . . . . . . . . . . . . . . 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)
224	172, 173, 175, 223	fmptapd 7025	. . . . . . . . . . . . . . . . . . 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))))
225	224	uneq1d 4092	. . . . . . . . . . . . . . . . . 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})))
226	171, 225	eqtrd 2778	. . . . . . . . . . . . . . . . 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})))
227		elmapfn 8611	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1st ‘𝑘) ∈ ((0..^𝐾) ↑m (1...(𝑀 + 1))) → (1st ‘𝑘) Fn (1...(𝑀 + 1)))
228	61, 227	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)))
229		fnssres 6539	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((1st ‘𝑘) Fn (1...(𝑀 + 1)) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((1st ‘𝑘) ↾ (1...𝑀)) Fn (1...𝑀))
230	228, 64, 229	sylancl 585	. . . . . . . . . . . . . . . . . . . . 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))
231	230	ad3antlr 727	. . . . . . . . . . . . . . . . . . . 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))
232		fnconstg 6646	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0 ∈ V → (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)))
233	163, 232	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))
234	210, 233	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((2nd ‘𝑘) “ (1...𝑗)) × {1}) Fn ((2nd ‘𝑘) “ (1...𝑗)) ∧ (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)))
235		imain 6503	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (Fun ◡(2nd ‘𝑘) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (((2nd ‘𝑘) “ (1...𝑗)) ∩ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀))))
236	76, 83, 235	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))))
237		fzdisj 13212	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
238	185, 237	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
239	238	imaeq2d 5958	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((2nd ‘𝑘) “ ∅))
240	239, 189	eqtrdi 2795	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ∅)
241	236, 240	sylan9req 2800	. . . . . . . . . . . . . . . . . . . . . . . 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))) = ∅)
242		fnun 6529	. . . . . . . . . . . . . . . . . . . . . . . 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))))
243	234, 241, 242	sylancr 586	. . . . . . . . . . . . . . . . . . . . . . 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))))
244	243	ad4ant24 750	. . . . . . . . . . . . . . . . . . . . . 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))))
245	101	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 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)}))
246	85	ad3antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . 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)})))
247	183, 194	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℕ)
248	247, 195	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ (ℤ≥‘1))
249		fzsplit2 13210	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑗 + 1) ∈ (ℤ≥‘1) ∧ 𝑀 ∈ (ℤ≥‘𝑗)) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
250	248, 199, 249	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑀) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
251	128, 250	sylan9eq 2799	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
252	251	imaeq2d 5958	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
253	246, 252	eqtr3d 2780	. . . . . . . . . . . . . . . . . . . . . . . . 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))))
254	125	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 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))
255	245, 253, 254	3eqtr3rd 2787	. . . . . . . . . . . . . . . . . . . . . . . 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))))
256		imaundi 6042	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)))
257	255, 256	eqtrdi 2795	. . . . . . . . . . . . . . . . . . . . . . 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))))
258	257	fneq2d 6511	. . . . . . . . . . . . . . . . . . . . . 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)))))
259	244, 258	mpbird 256	. . . . . . . . . . . . . . . . . . . . 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))
260		fzss2 13225	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑀 ∈ (ℤ≥‘𝑗) → (1...𝑗) ⊆ (1...𝑀))
261		resima2 5915	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1...𝑗) ⊆ (1...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) = ((2nd ‘𝑘) “ (1...𝑗)))
262	199, 260, 261	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) = ((2nd ‘𝑘) “ (1...𝑗)))
263	262	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) = (((2nd ‘𝑘) “ (1...𝑗)) × {1}))
264	183, 196	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ (ℤ≥‘1))
265		fzss1 13224	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑗 + 1) ∈ (ℤ≥‘1) → ((𝑗 + 1)...𝑀) ⊆ (1...𝑀))
266		resima2 5915	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑗 + 1)...𝑀) ⊆ (1...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)))
267	264, 265, 266	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)))
268	267	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0...𝑀) → ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))
269	263, 268	uneq12d 4094	. . . . . . . . . . . . . . . . . . . . . . 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})))
270	269	adantl 481	. . . . . . . . . . . . . . . . . . . . . 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})))
271	270	fneq1d 6510	. . . . . . . . . . . . . . . . . . . . 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)))
272	259, 271	mpbird 256	. . . . . . . . . . . . . . . . . . . 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))
273		fzfid 13621	. . . . . . . . . . . . . . . . . . . 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)
274		inidm 4149	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1...𝑀) ∩ (1...𝑀)) = (1...𝑀)
275		fvres 6775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑀) → (((1st ‘𝑘) ↾ (1...𝑀))‘𝑛) = ((1st ‘𝑘)‘𝑛))
276	275	adantl 481	. . . . . . . . . . . . . . . . . . . 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))
277		disjsn 4644	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ↔ ¬ (𝑀 + 1) ∈ (1...𝑀))
278	122, 277	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)
279	278	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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)}) = ∅)
280	259, 279	jca 511	. . . . . . . . . . . . . . . . . . . . . 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)}) = ∅))
281		fnconstg 6646	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 ∈ V → ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)})
282	163, 281	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)}
283		fvun1 6841	. . . . . . . . . . . . . . . . . . . . . . . 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))
284	282, 283	mp3an2 1447	. . . . . . . . . . . . . . . . . . . . . . 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))
285	284	anassrs 467	. . . . . . . . . . . . . . . . . . . . . 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))
286	280, 285	sylan 579	. . . . . . . . . . . . . . . . . . . . 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))
287	247	nnzd 12354	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℤ)
288	183	nn0cnd 12225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ)
289		pncan1 11329	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑗 ∈ ℂ → ((𝑗 + 1) − 1) = 𝑗)
290	288, 289	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1) − 1) = 𝑗)
291	290	fveq2d 6760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑗 ∈ (0...𝑀) → (ℤ≥‘((𝑗 + 1) − 1)) = (ℤ≥‘𝑗))
292	199, 291	eleqtrrd 2842	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ≥‘((𝑗 + 1) − 1)))
293		fzsuc2 13243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑗 + 1) ∈ ℤ ∧ 𝑀 ∈ (ℤ≥‘((𝑗 + 1) − 1))) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)}))
294	287, 292, 293	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)}))
295	294	imaeq2d 5958	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd ‘𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})))
296		imaundi 6042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((2nd ‘𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})) = (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)}))
297	295, 296	eqtrdi 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = (((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})))
298	297	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})) × {0}))
299		xpundir 5647	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd ‘𝑘) “ {(𝑀 + 1)})) × {0}) = ((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0}))
300	298, 299	eqtrdi 2795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd ‘𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd ‘𝑘) “ {(𝑀 + 1)}) × {0})))
301	300	uneq2d 4093	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))))
302		unass 4096	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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})))
303	301, 302	eqtr4di 2797	. . . . . . . . . . . . . . . . . . . . . . . . . . 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})))
304	303	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 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})))
305	98	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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}))
306	305	uneq2d 4093	. . . . . . . . . . . . . . . . . . . . . . . . . . 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})))
307	306	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 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})))
308	304, 307	eqtr4d 2781	. . . . . . . . . . . . . . . . . . . . . . . . 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})))
309	99	xpeq1d 5609	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → ({((2nd ‘𝑘)‘(𝑀 + 1))} × {0}) = ({(𝑀 + 1)} × {0}))
310	309	uneq2d 4093	. . . . . . . . . . . . . . . . . . . . . . . . 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})))
311	308, 310	sylan9eq 2799	. . . . . . . . . . . . . . . . . . . . . . . 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})))
312	311	an32s 648	. . . . . . . . . . . . . . . . . . . . . . 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})))
313	312	fveq1d 6758	. . . . . . . . . . . . . . . . . . . . . 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))
314	313	adantr 480	. . . . . . . . . . . . . . . . . . . . 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))
315	269	fveq1d 6758	. . . . . . . . . . . . . . . . . . . . . 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))
316	315	ad2antlr 723	. . . . . . . . . . . . . . . . . . . . 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))
317	286, 314, 316	3eqtr4rd 2789	. . . . . . . . . . . . . . . . . . . 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))
318	231, 272, 273, 273, 274, 276, 317	offval 7520	. . . . . . . . . . . . . . . . . . 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))))
319	318	uneq1d 4092	. . . . . . . . . . . . . . . . . 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})))
320	319	adantlrl 716	. . . . . . . . . . . . . . . . 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})))
321	228	adantr 480	. . . . . . . . . . . . . . . . . . . 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)))
322	210, 212	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . 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))))
323	181, 190	sylan9req 2800	. . . . . . . . . . . . . . . . . . . . . 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)))) = ∅)
324		fnun 6529	. . . . . . . . . . . . . . . . . . . . . 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)))))
325	322, 323, 324	sylancr 586	. . . . . . . . . . . . . . . . . . . . 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)))))
326		peano2uz 12570	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑀 ∈ (ℤ≥‘𝑗) → (𝑀 + 1) ∈ (ℤ≥‘𝑗))
327	199, 326	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑗))
328		fzsplit2 13210	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑗 + 1) ∈ (ℤ≥‘1) ∧ (𝑀 + 1) ∈ (ℤ≥‘𝑗)) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
329	264, 327, 328	syl2anc 583	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑀) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
330	329	imaeq2d 5958	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0...𝑀) → ((2nd ‘𝑘) “ (1...(𝑀 + 1))) = ((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))))
331		imaundi 6042	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((2nd ‘𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
332	330, 331	eqtr2di 2796	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ (0...𝑀) → (((2nd ‘𝑘) “ (1...𝑗)) ∪ ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd ‘𝑘) “ (1...(𝑀 + 1))))
333	332, 89	sylan9eqr 2801	. . . . . . . . . . . . . . . . . . . . . 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)))
334	333	fneq2d 6511	. . . . . . . . . . . . . . . . . . . . 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))))
335	325, 334	mpbid 231	. . . . . . . . . . . . . . . . . . . 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)))
336		fzfid 13621	. . . . . . . . . . . . . . . . . . . 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)
337		inidm 4149	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1...(𝑀 + 1)) ∩ (1...(𝑀 + 1))) = (1...(𝑀 + 1))
338		eqidd 2739	. . . . . . . . . . . . . . . . . . . 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))
339		eqidd 2739	. . . . . . . . . . . . . . . . . . . 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))
340	321, 335, 336, 336, 337, 338, 339	offval 7520	. . . . . . . . . . . . . . . . . . 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))))
341	340	uneq1d 4092	. . . . . . . . . . . . . . . . . 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})))
342	341	ad4ant24 750	. . . . . . . . . . . . . . . . 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})))
343	226, 320, 342	3eqtr4rd 2789	. . . . . . . . . . . . . . . 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})))
344	343	csbeq1d 3832	. . . . . . . . . . . . . . 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)
345	344	eqeq2d 2749	. . . . . . . . . . . . . 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))
346	345	rexbidva 3224	. . . . . . . . . . . . 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))
347	346	ralbidv 3120	. . . . . . . . . . . 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))
348	347	biimpd 228	. . . . . . . . . . 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))
349	348	impr 454	. . . . . . . . . 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)
350	139, 349	sylan2b 593	. . . . . . . . 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)
351		1st2nd2 7843	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → 𝑘 = ⟨(1st ‘𝑘), (2nd ‘𝑘)⟩)
352	351	ad2antlr 723	. . . . . . . . . . 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)⟩)
353		fnsnsplit 7038	. . . . . . . . . . . . . . . 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))⟩}))
354	228, 96, 353	syl2anr 596	. . . . . . . . . . . . . . 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))⟩}))
355	354	adantr 480	. . . . . . . . . . . . . 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))⟩}))
356	125	reseq2d 5880	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1st ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1st ‘𝑘) ↾ (1...𝑀)))
357	356	adantr 480	. . . . . . . . . . . . . . 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)))
358		opeq2 4802	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 → ⟨(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))⟩ = ⟨(𝑀 + 1), 0⟩)
359	358	sneqd 4570	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑘)‘(𝑀 + 1)) = 0 → {⟨(𝑀 + 1), ((1st ‘𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), 0⟩})
360		uneq12 4088	. . . . . . . . . . . . . . 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⟩}))
361	357, 359, 360	syl2an 595	. . . . . . . . . . . . . 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⟩}))
362	355, 361	eqtrd 2778	. . . . . . . . . . . . 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⟩}))
363	362	adantrr 713	. . . . . . . . . . . 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⟩}))
364		fnsnsplit 7038	. . . . . . . . . . . . . . . 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))⟩}))
365	92, 96, 364	syl2anr 596	. . . . . . . . . . . . . . 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))⟩}))
366	365	adantr 480	. . . . . . . . . . . . . 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))⟩}))
367	125	reseq2d 5880	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd ‘𝑘) ↾ (1...𝑀)))
368	367	adantr 480	. . . . . . . . . . . . . . 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)))
369		opeq2 4802	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → ⟨(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))⟩ = ⟨(𝑀 + 1), (𝑀 + 1)⟩)
370	369	sneqd 4570	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {⟨(𝑀 + 1), ((2nd ‘𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), (𝑀 + 1)⟩})
371		uneq12 4088	. . . . . . . . . . . . . . 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)⟩}))
372	368, 370, 371	syl2an 595	. . . . . . . . . . . . . 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)⟩}))
373	366, 372	eqtrd 2778	. . . . . . . . . . . . 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)⟩}))
374	373	adantrl 712	. . . . . . . . . . . 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)⟩}))
375	363, 374	opeq12d 4809	. . . . . . . . . . 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)⟩})⟩)
376	352, 375	eqtrd 2778	. . . . . . . . . 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)⟩})⟩)
377	376	3adantr1 1167	. . . . . . . . 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)⟩})⟩)
378		fvex 6769	. . . . . . . . . . . . . . . . . . 19 ⊢ (1st ‘𝑘) ∈ V
379	378	resex 5928	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑘) ↾ (1...𝑀)) ∈ V
380	379, 132	op1std 7814	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (1st ‘𝑡) = ((1st ‘𝑘) ↾ (1...𝑀)))
381	379, 132	op2ndd 7815	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (2nd ‘𝑡) = ((2nd ‘𝑘) ↾ (1...𝑀)))
382	381	imaeq1d 5957	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ((2nd ‘𝑡) “ (1...𝑗)) = (((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)))
383	382	xpeq1d 5609	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (((2nd ‘𝑡) “ (1...𝑗)) × {1}) = ((((2nd ‘𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}))
384	381	imaeq1d 5957	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) = (((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)))
385	384	xpeq1d 5609	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → (((2nd ‘𝑡) “ ((𝑗 + 1)...𝑀)) × {0}) = ((((2nd ‘𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))
386	383, 385	uneq12d 4094	. . . . . . . . . . . . . . . . 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})))
387	380, 386	oveq12d 7273	. . . . . . . . . . . . . . . 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}))))
388	387	uneq1d 4092	. . . . . . . . . . . . . . 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})))
389	388	csbeq1d 3832	. . . . . . . . . . . . . 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)
390	389	eqeq2d 2749	. . . . . . . . . . . . 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))
391	390	rexbidv 3225	. . . . . . . . . . . 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))
392	391	ralbidv 3120	. . . . . . . . . . 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))
393	380	uneq1d 4092	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = (((1st ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
394	381	uneq1d 4092	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨((1st ‘𝑘) ↾ (1...𝑀)), ((2nd ‘𝑘) ↾ (1...𝑀))⟩ → ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (((2nd ‘𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
395	393, 394	opeq12d 4809	. . . . . . . . . . . 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)⟩})⟩)
396	395	eqeq2d 2749	. . . . . . . . . . 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)⟩})⟩))
397	392, 396	anbi12d 630	. . . . . . . . . 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)⟩})⟩)))
398	397	rspcev 3552	. . . . . . . . 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)⟩})⟩))
399	137, 350, 377, 398	syl12anc 833	. . . . . . . 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)⟩})⟩))
400	399	ex 412	. . . . . . 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)⟩})⟩)))
401		elrabi 3611	. . . . . . . . . . 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)}))
402		elrabi 3611	. . . . . . . . . . 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)}))
403	401, 402	anim12i 612	. . . . . . . . . 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)})))
404		eqtr2 2762	. . . . . . . . . . . 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)⟩})⟩)
405	22, 24	opth 5385	. . . . . . . . . . . . 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)⟩})))
406		difeq1 4046	. . . . . . . . . . . . . . 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⟩}))
407		difun2 4411	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩})
408		difun2 4411	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩})
409	406, 407, 408	3eqtr3g 2802	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) → ((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}))
410		difeq1 4046	. . . . . . . . . . . . . . 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)⟩}))
411		difun2 4411	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})
412		difun2 4411	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})
413	410, 411, 412	3eqtr3g 2802	. . . . . . . . . . . . . 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)⟩}))
414	409, 413	anim12i 612	. . . . . . . . . . . . 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)⟩})))
415	405, 414	sylbi 216	. . . . . . . . . . . 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)⟩})))
416	404, 415	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)⟩})))
417		elmapfn 8611	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑡) ∈ ((0..^𝐾) ↑m (1...𝑀)) → (1st ‘𝑡) Fn (1...𝑀))
418		fnop 6526	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1st ‘𝑡) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑡)) → (𝑀 + 1) ∈ (1...𝑀))
419	418	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑡) Fn (1...𝑀) → (⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
420	9, 417, 419	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)))
421	420, 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)))
422	421	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑡))
423		difsn 4728	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑡) → ((1st ‘𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st ‘𝑡))
424	422, 423	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))
425		xp1st 7836	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st ‘𝑛) ∈ ((0..^𝐾) ↑m (1...𝑀)))
426		elmapfn 8611	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑛) ∈ ((0..^𝐾) ↑m (1...𝑀)) → (1st ‘𝑛) Fn (1...𝑀))
427		fnop 6526	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1st ‘𝑛) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑛)) → (𝑀 + 1) ∈ (1...𝑀))
428	427	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1st ‘𝑛) Fn (1...𝑀) → (⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
429	425, 426, 428	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)))
430	429, 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)))
431	430	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑛))
432		difsn 4728	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st ‘𝑛) → ((1st ‘𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st ‘𝑛))
433	431, 432	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))
434	424, 433	eqeqan12d 2752	. . . . . . . . . . . . . 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)))
435	434	anandis 674	. . . . . . . . . . . . 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)))
436		f1ofn 6701	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑡):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd ‘𝑡) Fn (1...𝑀))
437		fnop 6526	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘𝑡) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑡)) → (𝑀 + 1) ∈ (1...𝑀))
438	437	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑡) Fn (1...𝑀) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
439	17, 436, 438	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)))
440	439, 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)))
441	440	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑡))
442		difsn 4728	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑡) → ((2nd ‘𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd ‘𝑡))
443	441, 442	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))
444		xp2nd 7837	. . . . . . . . . . . . . . . . . . . 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)})
445		fvex 6769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2nd ‘𝑛) ∈ V
446		f1oeq1 6688	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = (2nd ‘𝑛) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀)))
447	445, 446	elab 3602	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘𝑛) ∈ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ (2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀))
448	444, 447	sylib 217	. . . . . . . . . . . . . . . . . . 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))
449		f1ofn 6701	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑛):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd ‘𝑛) Fn (1...𝑀))
450		fnop 6526	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((2nd ‘𝑛) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑛)) → (𝑀 + 1) ∈ (1...𝑀))
451	450	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑛) Fn (1...𝑀) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
452	448, 449, 451	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)))
453	452, 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)))
454	453	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑛))
455		difsn 4728	. . . . . . . . . . . . . . . 16 ⊢ (¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd ‘𝑛) → ((2nd ‘𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd ‘𝑛))
456	454, 455	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))
457	443, 456	eqeqan12d 2752	. . . . . . . . . . . . . 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)))
458	457	anandis 674	. . . . . . . . . . . . 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)))
459	435, 458	anbi12d 630	. . . . . . . . . . . 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))))
460		xpopth 7845	. . . . . . . . . . . . 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))
461	460	adantl 481	. . . . . . . . . . . 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))
462	459, 461	bitrd 278	. . . . . . . . . . 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))
463	416, 462	syl5ib 243	. . . . . . . . . 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))
464	403, 463	sylan2 592	. . . . . . . . 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))
465	464	ralrimivva 3114	. . . . . . . 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))
466	465	adantr 480	. . . . . . 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))
467	400, 466	jctird 526	. . . . . 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))))
468		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑛 → (1st ‘𝑡) = (1st ‘𝑛))
469	468	uneq1d 4092	. . . . . . . . . 10 ⊢ (𝑡 = 𝑛 → ((1st ‘𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st ‘𝑛) ∪ {⟨(𝑀 + 1), 0⟩}))
470		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑛 → (2nd ‘𝑡) = (2nd ‘𝑛))
471	470	uneq1d 4092	. . . . . . . . . 10 ⊢ (𝑡 = 𝑛 → ((2nd ‘𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd ‘𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
472	469, 471	opeq12d 4809	. . . . . . . . 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)⟩})⟩)
473	472	eqeq2d 2749	. . . . . . . 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)⟩})⟩))
474	473	reu4 3661	. . . . . . 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)))
475	58	rexrab 3626	. . . . . . . 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)⟩})⟩))
476	475	anbi1i 623	. . . . . . 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)))
477	474, 476	bitri 274	. . . . . 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)))
478	467, 477	syl6ibr 251	. . . . 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)⟩})⟩))
479	478	ralrimiva 3107	. . . 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)⟩})⟩))
480		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑘 → (1st ‘𝑠) = (1st ‘𝑘))
481		fveq2 6756	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 𝑘 → (2nd ‘𝑠) = (2nd ‘𝑘))
482	481	imaeq1d 5957	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd ‘𝑘) “ (1...𝑗)))
483	482	xpeq1d 5609	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = (((2nd ‘𝑘) “ (1...𝑗)) × {1}))
484	481	imaeq1d 5957	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
485	484	xpeq1d 5609	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((2nd ‘𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
486	483, 485	uneq12d 4094	. . . . . . . . . . . 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})))
487	480, 486	oveq12d 7273	. . . . . . . . . . 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}))))
488	487	uneq1d 4092	. . . . . . . . . 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})))
489	488	csbeq1d 3832	. . . . . . . . 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)
490	489	eqeq2d 2749	. . . . . . . 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))
491	490	rexbidv 3225	. . . . . . 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))
492	491	ralbidv 3120	. . . . . 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))
493	480	fveq1d 6758	. . . . . . 7 ⊢ (𝑠 = 𝑘 → ((1st ‘𝑠)‘(𝑀 + 1)) = ((1st ‘𝑘)‘(𝑀 + 1)))
494	493	eqeq1d 2740	. . . . . 6 ⊢ (𝑠 = 𝑘 → (((1st ‘𝑠)‘(𝑀 + 1)) = 0 ↔ ((1st ‘𝑘)‘(𝑀 + 1)) = 0))
495	481	fveq1d 6758	. . . . . . 7 ⊢ (𝑠 = 𝑘 → ((2nd ‘𝑠)‘(𝑀 + 1)) = ((2nd ‘𝑘)‘(𝑀 + 1)))
496	495	eqeq1d 2740	. . . . . 6 ⊢ (𝑠 = 𝑘 → (((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1) ↔ ((2nd ‘𝑘)‘(𝑀 + 1)) = (𝑀 + 1)))
497	492, 494, 496	3anbi123d 1434	. . . . 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))))
498	497	ralrab 3623	. . . 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)⟩})⟩))
499	479, 498	sylibr 233	. . 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)⟩})⟩)
500		eqid 2738	. . . 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)⟩})⟩)
501	500	f1ompt 6967	. . 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)⟩})⟩))
502	60, 499, 501	sylanbrc 582	. 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))})
503		ovex 7288	. . . . 5 ⊢ ((0..^𝐾) ↑m (1...𝑀)) ∈ V
504		ovex 7288	. . . . . 6 ⊢ ((1...𝑀) ↑m (1...𝑀)) ∈ V
505		f1of 6700	. . . . . . . 8 ⊢ (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑓:(1...𝑀)⟶(1...𝑀))
506	505	ss2abi 3996	. . . . . . 7 ⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ {𝑓 ∣ 𝑓:(1...𝑀)⟶(1...𝑀)}
507	68, 68	mapval 8585	. . . . . . 7 ⊢ ((1...𝑀) ↑m (1...𝑀)) = {𝑓 ∣ 𝑓:(1...𝑀)⟶(1...𝑀)}
508	506, 507	sseqtrri 3954	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ ((1...𝑀) ↑m (1...𝑀))
509	504, 508	ssexi 5241	. . . . 5 ⊢ {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ∈ V
510	503, 509	xpex 7581	. . . 4 ⊢ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∈ V
511	510	rabex 5251	. . 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
512	511	f1oen 8716	. 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))})
513	502, 512	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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  ∃!wreu 3065  {crab 3067  Vcvv 3422  ⦋csb 3828   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  ^◡ccnv 5579   ↾ cres 5582   “ cima 5583  Fun wfun 6412   Fn wfn 6413  ⟶wf 6414  –1-1→wf1 6415  –onto→wfo 6416  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509  1^st c1st 7802  2^nd c2nd 7803   ↑_m cmap 8573   ≈ cen 8688  Fincfn 8691  ℂcc 10800  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805   < clt 10940   ≤ cle 10941   − cmin 11135  ℕcn 11903  ℕ₀cn0 12163  ℤcz 12249  ℤ_≥cuz 12511  ...cfz 13168  ..^cfzo 13311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312

This theorem is referenced by:  poimirlem28  35732