poimirlem3 - Mathbox for Brendan Leahy

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Theorem poimirlem3 33044

Description: Lemma for poimir 33074 to add an interior point to an admissible face on the back face of the 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)
poimirlem3.4	⊢ (𝜑 → 𝑇:(1...𝑀)⟶(0..^𝐾))
poimirlem3.5	⊢ (𝜑 → 𝑈:(1...𝑀)–1-1-onto→(1...𝑀))

**Assertion**
Ref	Expression
poimirlem3	⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(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 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))

Distinct variable groups: 𝑓,𝑖,𝑗,𝑝 𝜑,𝑗 𝑗,𝑀 𝑗,𝑁 𝑇,𝑗 𝑈,𝑗 𝜑,𝑖,𝑝 𝐵,𝑓,𝑖,𝑗 𝑓,𝐾,𝑖,𝑗,𝑝 𝑓,𝑀,𝑖,𝑝 𝑓,𝑁,𝑖,𝑝 𝑇,𝑓,𝑖,𝑝 𝑈,𝑓,𝑖,𝑝 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑝)

Proof of Theorem poimirlem3 Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimirlem3.4	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇:(1...𝑀)⟶(0..^𝐾))
2		ffn 6002	. . . . . . . . . . . . . . 15 ⊢ (𝑇:(1...𝑀)⟶(0..^𝐾) → 𝑇 Fn (1...𝑀))
3	1, 2	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 Fn (1...𝑀))
4	3	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 𝑇 Fn (1...𝑀))
5		1ex 9979	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
6		fnconstg 6050	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ V → ((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)))
7	5, 6	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗))
8		c0ex 9978	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
9		fnconstg 6050	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ V → ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀)))
10	8, 9	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀))
11	7, 10	pm3.2i 471	. . . . . . . . . . . . . . 15 ⊢ (((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)) ∧ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀)))
12		poimirlem3.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈:(1...𝑀)–1-1-onto→(1...𝑀))
13		dff1o3 6100	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑈:(1...𝑀)–onto→(1...𝑀) ∧ Fun ^◡𝑈))
14	13	simprbi 480	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → Fun ^◡𝑈)
15		imain 5932	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ^◡𝑈 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))))
16	12, 14, 15	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))))
17		elfznn0 12374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ₀)
18	17	nn0red 11296	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
19	18	ltp1d 10898	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → 𝑗 < (𝑗 + 1))
20		fzdisj 12310	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
21	19, 20	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
22	21	imaeq2d 5425	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (𝑈 “ ∅))
23		ima0 5440	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 “ ∅) = ∅
24	22, 23	syl6eq 2671	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑀) → (𝑈 “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ∅)
25	16, 24	sylan9req 2676	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))) = ∅)
26		fnun 5955	. . . . . . . . . . . . . . 15 ⊢ (((((𝑈 “ (1...𝑗)) × {1}) Fn (𝑈 “ (1...𝑗)) ∧ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) Fn (𝑈 “ ((𝑗 + 1)...𝑀))) ∧ ((𝑈 “ (1...𝑗)) ∩ (𝑈 “ ((𝑗 + 1)...𝑀))) = ∅) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))))
27	11, 25, 26	sylancr 694	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))))
28		imaundi 5504	. . . . . . . . . . . . . . . 16 ⊢ (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀)))
29		nn0p1nn 11276	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ ℕ)
30		nnuz 11667	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ = (ℤ_≥‘1)
31	29, 30	syl6eleq 2708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℕ₀ → (𝑗 + 1) ∈ (ℤ_≥‘1))
32	17, 31	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ (ℤ_≥‘1))
33		elfzuz3 12281	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ_≥‘𝑗))
34		fzsplit2 12308	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ (ℤ_≥‘𝑗)) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
35	32, 33, 34	syl2anc 692	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
36	35	eqcomd 2627	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)) = (1...𝑀))
37	36	imaeq2d 5425	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (𝑈 “ (1...𝑀)))
38		f1ofo 6101	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑈:(1...𝑀)–onto→(1...𝑀))
39		foima 6077	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑈:(1...𝑀)–onto→(1...𝑀) → (𝑈 “ (1...𝑀)) = (1...𝑀))
40	12, 38, 39	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑈 “ (1...𝑀)) = (1...𝑀))
41	37, 40	sylan9eqr 2677	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑈 “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (1...𝑀))
42	28, 41	syl5eqr 2669	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))) = (1...𝑀))
43	42	fneq2d 5940	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn ((𝑈 “ (1...𝑗)) ∪ (𝑈 “ ((𝑗 + 1)...𝑀))) ↔ (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀)))
44	27, 43	mpbid 222	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀))
45		ovex 6632	. . . . . . . . . . . . . 14 ⊢ (1...𝑀) ∈ V
46	45	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) ∈ V)
47		inidm 3800	. . . . . . . . . . . . 13 ⊢ ((1...𝑀) ∩ (1...𝑀)) = (1...𝑀)
48		eqidd 2622	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (𝑇‘𝑛) = (𝑇‘𝑛))
49		eqidd 2622	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
50	4, 44, 46, 46, 47, 48, 49	offval 6857	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) = (𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))))
51		poimirlem4.2	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
52		nn0p1nn 11276	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ ℕ₀ → (𝑀 + 1) ∈ ℕ)
53	51, 52	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
54	53	nnzd 11425	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
55		uzid 11646	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 + 1) ∈ ℤ → (𝑀 + 1) ∈ (ℤ_≥‘(𝑀 + 1)))
56		peano2uz 11685	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘(𝑀 + 1)) → ((𝑀 + 1) + 1) ∈ (ℤ_≥‘(𝑀 + 1)))
57	54, 55, 56	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑀 + 1) + 1) ∈ (ℤ_≥‘(𝑀 + 1)))
58		poimirlem4.3	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑀 < 𝑁)
59	51	nn0zd 11424	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ ℤ)
60		poimir.0	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℕ)
61	60	nnzd 11425	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
62		zltp1le 11371	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
63		peano2z 11362	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ)
64		eluz 11645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
65	63, 64	sylan 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ_≥‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
66	62, 65	bitr4d 271	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ_≥‘(𝑀 + 1))))
67	59, 61, 66	syl2anc 692	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑁 ∈ (ℤ_≥‘(𝑀 + 1))))
68	58, 67	mpbid 222	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑀 + 1)))
69		fzsplit2 12308	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑀 + 1) + 1) ∈ (ℤ_≥‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ_≥‘(𝑀 + 1))) → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
70	57, 68, 69	syl2anc 692	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
71		fzsn 12325	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 + 1) ∈ ℤ → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
72	54, 71	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
73	72	uneq1d 3744	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
74	70, 73	eqtrd 2655	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
75	74	xpeq1d 5098	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}))
76		xpundir 5133	. . . . . . . . . . . . . . 15 ⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
77		ovex 6632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 + 1) ∈ V
78	77, 8	xpsn 6361	. . . . . . . . . . . . . . . 16 ⊢ ({(𝑀 + 1)} × {0}) = {⟨(𝑀 + 1), 0⟩}
79	78	uneq1i 3741	. . . . . . . . . . . . . . 15 ⊢ (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
80	76, 79	eqtri 2643	. . . . . . . . . . . . . 14 ⊢ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
81	75, 80	syl6eq 2671	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
82	81	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑀 + 1)...𝑁) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
83	50, 82	uneq12d 3746	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))))
84		unass 3748	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
85	83, 84	syl6eqr 2673	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
86	51	nn0red 11296	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑀 ∈ ℝ)
87	86	ltp1d 10898	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
88	53	nnred 10979	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
89	86, 88	ltnled 10128	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
90	87, 89	mpbid 222	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
91		elfzle2 12287	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 + 1) ∈ (1...𝑀) → (𝑀 + 1) ≤ 𝑀)
92	90, 91	nsyl 135	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ¬ (𝑀 + 1) ∈ (1...𝑀))
93		disjsn 4216	. . . . . . . . . . . . . . . . . 18 ⊢ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ↔ ¬ (𝑀 + 1) ∈ (1...𝑀))
94	92, 93	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)
95		eqid 2621	. . . . . . . . . . . . . . . . . . 19 ⊢ {⟨(𝑀 + 1), 0⟩} = {⟨(𝑀 + 1), 0⟩}
96	77, 8	fsn 6356	. . . . . . . . . . . . . . . . . . 19 ⊢ ({⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0} ↔ {⟨(𝑀 + 1), 0⟩} = {⟨(𝑀 + 1), 0⟩})
97	95, 96	mpbir 221	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0}
98		fun 6023	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑇:(1...𝑀)⟶(0..^𝐾) ∧ {⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0}) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
99	97, 98	mpanl2 716	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑇:(1...𝑀)⟶(0..^𝐾) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
100	1, 94, 99	syl2anc 692	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
101		1z 11351	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℤ
102		nn0uz 11666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ₀ = (ℤ_≥‘0)
103		1m1e0 11033	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1 − 1) = 0
104	103	fveq2i 6151	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℤ_≥‘(1 − 1)) = (ℤ_≥‘0)
105	102, 104	eqtr4i 2646	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ₀ = (ℤ_≥‘(1 − 1))
106	51, 105	syl6eleq 2708	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(1 − 1)))
107		fzsuc2 12340	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ (ℤ_≥‘(1 − 1))) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
108	101, 106, 107	sylancr 694	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
109	108	eqcomd 2627	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
110		poimirlem4.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 ∈ ℕ)
111		lbfzo0 12448	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0 ∈ (0..^𝐾) ↔ 𝐾 ∈ ℕ)
112	110, 111	sylibr 224	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 ∈ (0..^𝐾))
113	112	snssd 4309	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {0} ⊆ (0..^𝐾))
114		ssequn2 3764	. . . . . . . . . . . . . . . . . 18 ⊢ ({0} ⊆ (0..^𝐾) ↔ ((0..^𝐾) ∪ {0}) = (0..^𝐾))
115	113, 114	sylib 208	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0..^𝐾) ∪ {0}) = (0..^𝐾))
116	109, 115	feq23d 5997	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}) ↔ (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾)))
117	100, 116	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾))
118		ffn 6002	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾) → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) Fn (1...(𝑀 + 1)))
119	117, 118	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) Fn (1...(𝑀 + 1)))
120	119	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) Fn (1...(𝑀 + 1)))
121		fnconstg 6050	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ V → (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)))
122	5, 121	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗))
123		fnconstg 6050	. . . . . . . . . . . . . . . . 17 ⊢ (0 ∈ V → (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
124	8, 123	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))
125	122, 124	pm3.2i 471	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
126	77, 77	f1osn 6133	. . . . . . . . . . . . . . . . . . 19 ⊢ {⟨(𝑀 + 1), (𝑀 + 1)⟩}:{(𝑀 + 1)}–1-1-onto→{(𝑀 + 1)}
127		f1oun 6113	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ∧ {⟨(𝑀 + 1), (𝑀 + 1)⟩}:{(𝑀 + 1)}–1-1-onto→{(𝑀 + 1)}) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}))
128	126, 127	mpanl2 716	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}))
129	12, 94, 94, 128	syl12anc 1321	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}))
130		dff1o3 6100	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}) ↔ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–onto→((1...𝑀) ∪ {(𝑀 + 1)}) ∧ Fun ^◡(𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
131	130	simprbi 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}) → Fun ^◡(𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
132		imain 5932	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ^◡(𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
133	129, 131, 132	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
134		fzdisj 12310	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
135	19, 134	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
136	135	imaeq2d 5425	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ∅))
137		ima0 5440	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ∅) = ∅
138	136, 137	syl6eq 2671	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
139	133, 138	sylan9req 2676	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
140		fnun 5955	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
141	125, 139, 140	sylancr 694	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
142		f1ofo 6101	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)}) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–onto→((1...𝑀) ∪ {(𝑀 + 1)}))
143		foima 6077	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–onto→((1...𝑀) ∪ {(𝑀 + 1)}) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑀) ∪ {(𝑀 + 1)})) = ((1...𝑀) ∪ {(𝑀 + 1)}))
144	129, 142, 143	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑀) ∪ {(𝑀 + 1)})) = ((1...𝑀) ∪ {(𝑀 + 1)}))
145	108	imaeq2d 5425	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...(𝑀 + 1))) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑀) ∪ {(𝑀 + 1)})))
146	144, 145, 108	3eqtr4d 2665	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
147		peano2uz 11685	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (ℤ_≥‘𝑗) → (𝑀 + 1) ∈ (ℤ_≥‘𝑗))
148	33, 147	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ (ℤ_≥‘𝑗))
149		fzsplit2 12308	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 + 1) ∈ (ℤ_≥‘1) ∧ (𝑀 + 1) ∈ (ℤ_≥‘𝑗)) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
150	32, 148, 149	syl2anc 692	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
151	150	imaeq2d 5425	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (0...𝑀) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...(𝑀 + 1))) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))))
152	146, 151	sylan9req 2676	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) = ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))))
153		imaundi 5504	. . . . . . . . . . . . . . . 16 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
154	152, 153	syl6eq 2671	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) = (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))))
155	154	fneq2d 5940	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1)) ↔ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∪ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))))
156	141, 155	mpbird 247	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1)))
157		ovex 6632	. . . . . . . . . . . . . 14 ⊢ (1...(𝑀 + 1)) ∈ V
158	157	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) ∈ V)
159		inidm 3800	. . . . . . . . . . . . 13 ⊢ ((1...(𝑀 + 1)) ∩ (1...(𝑀 + 1))) = (1...(𝑀 + 1))
160		eqidd 2622	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛))
161		eqidd 2622	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))
162	120, 156, 158, 158, 159, 160, 161	offval 6857	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
163	77	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ V)
164	8	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ V)
165	109	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
166		fveq2 6148	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑀 + 1) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)))
167	77	snid 4179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 + 1) ∈ {(𝑀 + 1)}
168	77, 8	fnsn 5904	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {⟨(𝑀 + 1), 0⟩} Fn {(𝑀 + 1)}
169		fvun2 6227	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑇 Fn (1...𝑀) ∧ {⟨(𝑀 + 1), 0⟩} Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
170	168, 169	mp3an2 1409	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑇 Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
171	167, 170	mpanr2 719	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
172	3, 94, 171	syl2anc 692	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)))
173	77, 8	fvsn 6400	. . . . . . . . . . . . . . . . . 18 ⊢ ({⟨(𝑀 + 1), 0⟩}‘(𝑀 + 1)) = 0
174	172, 173	syl6eq 2671	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0)
175	166, 174	sylan9eqr 2677	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 = (𝑀 + 1)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = 0)
176	175	adantlr 750	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = 0)
177		fveq2 6148	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑀 + 1) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)))
178		imadmrn 5435	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (({(𝑀 + 1)} × {(𝑀 + 1)}) “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ran ({(𝑀 + 1)} × {(𝑀 + 1)})
179	77, 77	xpsn 6361	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({(𝑀 + 1)} × {(𝑀 + 1)}) = {⟨(𝑀 + 1), (𝑀 + 1)⟩}
180	179	imaeq1i 5422	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (({(𝑀 + 1)} × {(𝑀 + 1)}) “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ dom ({(𝑀 + 1)} × {(𝑀 + 1)}))
181		dmxpid 5305	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ dom ({(𝑀 + 1)} × {(𝑀 + 1)}) = {(𝑀 + 1)}
182	181	imaeq2i 5423	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)})
183	180, 182	eqtri 2643	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (({(𝑀 + 1)} × {(𝑀 + 1)}) “ dom ({(𝑀 + 1)} × {(𝑀 + 1)})) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)})
184		rnxpid 5526	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ran ({(𝑀 + 1)} × {(𝑀 + 1)}) = {(𝑀 + 1)}
185	178, 183, 184	3eqtr3ri 2652	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {(𝑀 + 1)} = ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)})
186		eluzp1p1 11657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑀 ∈ (ℤ_≥‘𝑗) → (𝑀 + 1) ∈ (ℤ_≥‘(𝑗 + 1)))
187		eluzfz2 12291	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘(𝑗 + 1)) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
188	33, 186, 187	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
189	188	snssd 4309	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0...𝑀) → {(𝑀 + 1)} ⊆ ((𝑗 + 1)...(𝑀 + 1)))
190		imass2 5460	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({(𝑀 + 1)} ⊆ ((𝑗 + 1)...(𝑀 + 1)) → ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)}) ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
191	189, 190	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ (0...𝑀) → ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ {(𝑀 + 1)}) ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
192	185, 191	syl5eqss 3628	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ (0...𝑀) → {(𝑀 + 1)} ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
193		ssel 3577	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ({(𝑀 + 1)} ⊆ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) → ((𝑀 + 1) ∈ {(𝑀 + 1)} → (𝑀 + 1) ∈ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))))
194	192, 167, 193	mpisyl 21	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
195		elun2 3759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑀 + 1) ∈ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) → (𝑀 + 1) ∈ ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))))
196	194, 195	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))))
197		imaundir 5505	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) = ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
198	196, 197	syl6eleqr 2709	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
199	198	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
200		fvun2 6227	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∧ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) ∧ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
201	122, 124, 200	mp3an12 1411	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) ∩ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1)))) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
202	139, 199, 201	syl2anc 692	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
203	8	fvconst2 6423	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 + 1) ∈ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
204	198, 203	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (0...𝑀) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
205	204	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
206	202, 205	eqtrd 2655	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = 0)
207	177, 206	sylan9eqr 2677	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = 0)
208	176, 207	oveq12d 6622	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = (0 + 0))
209		00id 10155	. . . . . . . . . . . . . 14 ⊢ (0 + 0) = 0
210	208, 209	syl6eq 2671	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = 0)
211	163, 164, 165, 210	fmptapd 6391	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
212	3, 94	jca 554	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑇 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅))
213		fvun1 6226	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑇 Fn (1...𝑀) ∧ {⟨(𝑀 + 1), 0⟩} Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇‘𝑛))
214	168, 213	mp3an2 1409	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇 Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇‘𝑛))
215	214	anassrs 679	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) ∧ 𝑛 ∈ (1...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇‘𝑛))
216	212, 215	sylan 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇‘𝑛))
217	216	adantlr 750	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) = (𝑇‘𝑛))
218		fvres 6164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...𝑀) → ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))‘𝑛) = (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))
219	218	eqcomd 2627	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑀) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))‘𝑛))
220		resundir 5370	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀)) = (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)))
221		relxp 5188	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Rel ((𝑈 “ (1...𝑗)) × {1})
222		dmxpss 5524	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ (𝑈 “ (1...𝑗))
223		imassrn 5436	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑈 “ (1...𝑗)) ⊆ ran 𝑈
224	222, 223	sstri 3592	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ ran 𝑈
225		f1of 6094	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑈:(1...𝑀)⟶(1...𝑀))
226		frn 6010	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑈:(1...𝑀)⟶(1...𝑀) → ran 𝑈 ⊆ (1...𝑀))
227	12, 225, 226	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ran 𝑈 ⊆ (1...𝑀))
228	224, 227	syl5ss 3594	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ (1...𝑀))
229		relssres 5396	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Rel ((𝑈 “ (1...𝑗)) × {1}) ∧ dom ((𝑈 “ (1...𝑗)) × {1}) ⊆ (1...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
230	221, 228, 229	sylancr 694	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
231	230	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
232		imassrn 5436	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ⊆ ran {⟨(𝑀 + 1), (𝑀 + 1)⟩}
233	77	rnsnop 5575	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ran {⟨(𝑀 + 1), (𝑀 + 1)⟩} = {(𝑀 + 1)}
234	232, 233	sseqtri 3616	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ⊆ {(𝑀 + 1)}
235		ssrin 3816	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ⊆ {(𝑀 + 1)} → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀)))
236	234, 235	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀))
237		incom 3783	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({(𝑀 + 1)} ∩ (1...𝑀)) = ((1...𝑀) ∩ {(𝑀 + 1)})
238	237, 94	syl5eq 2667	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ({(𝑀 + 1)} ∩ (1...𝑀)) = ∅)
239	236, 238	syl5sseq 3632	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ∅)
240		ss0 3946	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) ⊆ ∅ → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅)
241	239, 240	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅)
242		fnconstg 6050	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 ∈ V → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)))
243		fnresdisj 5959	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅))
244	5, 242, 243	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅)
245	241, 244	sylib 208	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅)
246	245	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ∅)
247	231, 246	uneq12d 3746	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀))) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ∅))
248		imaundir 5505	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) = ((𝑈 “ (1...𝑗)) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)))
249	248	xpeq1i 5095	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) = (((𝑈 “ (1...𝑗)) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗))) × {1})
250		xpundir 5133	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑈 “ (1...𝑗)) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗))) × {1}) = (((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}))
251	249, 250	eqtri 2643	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) = (((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}))
252	251	reseq1i 5352	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1})) ↾ (1...𝑀))
253		resundir 5370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑈 “ (1...𝑗)) × {1}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1})) ↾ (1...𝑀)) = ((((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀)))
254	252, 253	eqtr2i 2644	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑈 “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ (1...𝑗)) × {1}) ↾ (1...𝑀))) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀))
255		un0 3939	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑈 “ (1...𝑗)) × {1}) ∪ ∅) = ((𝑈 “ (1...𝑗)) × {1})
256	247, 254, 255	3eqtr3g 2678	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) = ((𝑈 “ (1...𝑗)) × {1}))
257		f1odm 6098	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → dom 𝑈 = (1...𝑀))
258	12, 257	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → dom 𝑈 = (1...𝑀))
259	258	ineq2d 3792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (((𝑗 + 1)...(𝑀 + 1)) ∩ dom 𝑈) = (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀)))
260	259	reseq2d 5356	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ dom 𝑈)) = (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀))))
261		f1orel 6097	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → Rel 𝑈)
262		resindm 5403	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (Rel 𝑈 → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ dom 𝑈)) = (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))))
263	12, 261, 262	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ dom 𝑈)) = (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))))
264	260, 263	eqtr3d 2657	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀))) = (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))))
265	35	ineq2d 3792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀)) = (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
266		fzssp1 12326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑗 + 1)...𝑀) ⊆ ((𝑗 + 1)...(𝑀 + 1))
267		sseqin2 3795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑗 + 1)...𝑀) ⊆ ((𝑗 + 1)...(𝑀 + 1)) ↔ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) = ((𝑗 + 1)...𝑀))
268	266, 267	mpbi 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) = ((𝑗 + 1)...𝑀)
269	268	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) = ((𝑗 + 1)...𝑀))
270		incom 3783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) = ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))
271	270, 135	syl5eq 2667	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) = ∅)
272	269, 271	uneq12d 3746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 ∈ (0...𝑀) → ((((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗))) = (((𝑗 + 1)...𝑀) ∪ ∅))
273		uncom 3735	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗))) = ((((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)))
274		indi 3849	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = ((((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)))
275	273, 274	eqtr4i 2646	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑗 + 1)...(𝑀 + 1)) ∩ ((𝑗 + 1)...𝑀)) ∪ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑗))) = (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
276		un0 3939	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑗 + 1)...𝑀) ∪ ∅) = ((𝑗 + 1)...𝑀)
277	272, 275, 276	3eqtr3g 2678	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = ((𝑗 + 1)...𝑀))
278	265, 277	eqtrd 2655	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀)) = ((𝑗 + 1)...𝑀))
279	278	reseq2d 5356	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (0...𝑀) → (𝑈 ↾ (((𝑗 + 1)...(𝑀 + 1)) ∩ (1...𝑀))) = (𝑈 ↾ ((𝑗 + 1)...𝑀)))
280	264, 279	sylan9req 2676	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))) = (𝑈 ↾ ((𝑗 + 1)...𝑀)))
281	280	rneqd 5313	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ran (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1))) = ran (𝑈 ↾ ((𝑗 + 1)...𝑀)))
282		df-ima 5087	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) = ran (𝑈 ↾ ((𝑗 + 1)...(𝑀 + 1)))
283		df-ima 5087	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑈 “ ((𝑗 + 1)...𝑀)) = ran (𝑈 ↾ ((𝑗 + 1)...𝑀))
284	281, 282, 283	3eqtr4g 2680	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) = (𝑈 “ ((𝑗 + 1)...𝑀)))
285	284	xpeq1d 5098	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
286	285	reseq1d 5355	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)))
287		relxp 5188	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Rel ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})
288		dmxpss 5524	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ (𝑈 “ ((𝑗 + 1)...𝑀))
289		imassrn 5436	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑈 “ ((𝑗 + 1)...𝑀)) ⊆ ran 𝑈
290	288, 289	sstri 3592	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ ran 𝑈
291	290, 227	syl5ss 3594	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ (1...𝑀))
292		relssres 5396	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Rel ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ∧ dom ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ⊆ (1...𝑀)) → (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
293	287, 291, 292	sylancr 694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
294	293	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
295	286, 294	eqtrd 2655	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
296		imassrn 5436	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ⊆ ran {⟨(𝑀 + 1), (𝑀 + 1)⟩}
297	296, 233	sseqtri 3616	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ⊆ {(𝑀 + 1)}
298		ssrin 3816	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ⊆ {(𝑀 + 1)} → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀)))
299	297, 298	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ({(𝑀 + 1)} ∩ (1...𝑀))
300	299, 238	syl5sseq 3632	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ∅)
301		ss0 3946	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) ⊆ ∅ → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅)
302	300, 301	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅)
303		fnconstg 6050	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 ∈ V → (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))))
304		fnresdisj 5959	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅))
305	8, 303, 304	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) ∩ (1...𝑀)) = ∅ ↔ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅)
306	302, 305	sylib 208	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅)
307	306	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ∅)
308	295, 307	uneq12d 3746	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))) = (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ∪ ∅))
309	197	xpeq1i 5095	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))) × {0})
310		xpundir 5133	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) ∪ ({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1)))) × {0}) = (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
311	309, 310	eqtri 2643	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
312	311	reseq1i 5352	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))
313		resundir 5370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ∪ (({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀)) = ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)))
314	312, 313	eqtr2i 2644	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑈 “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) ∪ ((({⟨(𝑀 + 1), (𝑀 + 1)⟩} “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))) = ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))
315		un0 3939	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}) ∪ ∅) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})
316	308, 314, 315	3eqtr3g 2678	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀)) = ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))
317	256, 316	uneq12d 3746	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ↾ (1...𝑀)) ∪ ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) ↾ (1...𝑀))) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})))
318	220, 317	syl5eq 2667	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀)) = (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0})))
319	318	fveq1d 6150	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) ↾ (1...𝑀))‘𝑛) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
320	219, 319	sylan9eqr 2677	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
321	217, 320	oveq12d 6622	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛)))
322	321	mpteq2dva 4704	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑛 ∈ (1...𝑀) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) = (𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))))
323	322	uneq1d 3744	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘𝑛) + (((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}))
324	162, 211, 323	3eqtr2d 2661	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = ((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}))
325	324	uneq1d 3744	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ ((𝑇‘𝑛) + ((((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
326	85, 325	eqtr4d 2658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
327	326	csbeq1d 3521	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → ⦋((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
328	327	eqeq2d 2631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑀)) → (𝑖 = ⦋((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
329	328	rexbidva 3042	. . . . . 6 ⊢ (𝜑 → (∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
330	329	ralbidv 2980	. . . . 5 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
331	330	biimpd 219	. . . 4 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
332		f1ofn 6095	. . . . . . . 8 ⊢ (𝑈:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑈 Fn (1...𝑀))
333	12, 332	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 Fn (1...𝑀))
334	77, 77	fnsn 5904	. . . . . . . . 9 ⊢ {⟨(𝑀 + 1), (𝑀 + 1)⟩} Fn {(𝑀 + 1)}
335		fvun2 6227	. . . . . . . . 9 ⊢ ((𝑈 Fn (1...𝑀) ∧ {⟨(𝑀 + 1), (𝑀 + 1)⟩} Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
336	334, 335	mp3an2 1409	. . . . . . . 8 ⊢ ((𝑈 Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ (𝑀 + 1) ∈ {(𝑀 + 1)})) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
337	167, 336	mpanr2 719	. . . . . . 7 ⊢ ((𝑈 Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
338	333, 94, 337	syl2anc 692	. . . . . 6 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)))
339	77, 77	fvsn 6400	. . . . . 6 ⊢ ({⟨(𝑀 + 1), (𝑀 + 1)⟩}‘(𝑀 + 1)) = (𝑀 + 1)
340	338, 339	syl6eq 2671	. . . . 5 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))
341	174, 340	jca 554	. . . 4 ⊢ (𝜑 → (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))
342	331, 341	jctird 566	. . 3 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))
343		3anass 1040	. . 3 ⊢ ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
344	342, 343	syl6ibr 242	. 2 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
345	1, 97	jctir 560	. . . . . 6 ⊢ (𝜑 → (𝑇:(1...𝑀)⟶(0..^𝐾) ∧ {⟨(𝑀 + 1), 0⟩}:{(𝑀 + 1)}⟶{0}))
346	345, 94, 98	syl2anc 692	. . . . 5 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):((1...𝑀) ∪ {(𝑀 + 1)})⟶((0..^𝐾) ∪ {0}))
347	346, 116	mpbid 222	. . . 4 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾))
348		ovex 6632	. . . . 5 ⊢ (0..^𝐾) ∈ V
349	348, 157	elmap 7830	. . . 4 ⊢ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∈ ((0..^𝐾) ↑_𝑚 (1...(𝑀 + 1))) ↔ (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}):(1...(𝑀 + 1))⟶(0..^𝐾))
350	347, 349	sylibr 224	. . 3 ⊢ (𝜑 → (𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∈ ((0..^𝐾) ↑_𝑚 (1...(𝑀 + 1))))
351		f1oexrnex 7062	. . . . . . . 8 ⊢ ((𝑈:(1...𝑀)–1-1-onto→(1...𝑀) ∧ (1...𝑀) ∈ V) → 𝑈 ∈ V)
352	12, 45, 351	sylancl 693	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ V)
353		snex 4869	. . . . . . 7 ⊢ {⟨(𝑀 + 1), (𝑀 + 1)⟩} ∈ V
354		unexg 6912	. . . . . . 7 ⊢ ((𝑈 ∈ V ∧ {⟨(𝑀 + 1), (𝑀 + 1)⟩} ∈ V) → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V)
355	352, 353, 354	sylancl 693	. . . . . 6 ⊢ (𝜑 → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V)
356		f1oeq1 6084	. . . . . . 7 ⊢ (𝑓 = (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → (𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
357	356	elabg 3334	. . . . . 6 ⊢ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
358	355, 357	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
359		f1oeq23 6087	. . . . . 6 ⊢ (((1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}) ∧ (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)})) → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)})))
360	108, 108, 359	syl2anc 692	. . . . 5 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)})))
361	358, 360	bitrd 268	. . . 4 ⊢ (𝜑 → ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}):((1...𝑀) ∪ {(𝑀 + 1)})–1-1-onto→((1...𝑀) ∪ {(𝑀 + 1)})))
362	129, 361	mpbird 247	. . 3 ⊢ (𝜑 → (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})
363		opelxpi 5108	. . 3 ⊢ (((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∈ ((0..^𝐾) ↑_𝑚 (1...(𝑀 + 1))) ∧ (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑_𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}))
364	350, 362, 363	syl2anc 692	. 2 ⊢ (𝜑 → ⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑_𝑚 (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}))
365	344, 364	jctild 565	1 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(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 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘_𝑓 + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1036 = wceq 1480 ∈ wcel 1987 {cab 2607 ∀wral 2907 ∃wrex 2908 Vcvv 3186 ⦋csb 3514 ∪ cun 3553 ∩ cin 3554 ⊆ wss 3555 ∅c0 3891 {csn 4148 ⟨cop 4154 class class class wbr 4613 ↦ cmpt 4673 × cxp 5072 ^◡ccnv 5073 dom cdm 5074 ran crn 5075 ↾ cres 5076 “ cima 5077 Rel wrel 5079 Fun wfun 5841 Fn wfn 5842 ⟶wf 5843 –onto→wfo 5845 –1-1-onto→wf1o 5846 ‘cfv 5847 (class class class)co 6604 ∘_𝑓 cof 6848 ↑_𝑚 cmap 7802 0cc0 9880 1c1 9881 + caddc 9883 < clt 10018 ≤ cle 10019 − cmin 10210 ℕcn 10964 ℕ₀cn0 11236 ℤcz 11321 ℤ_≥cuz 11631 ...cfz 12268 ..^cfzo 12406

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1836 ax-6 1885 ax-7 1932 ax-8 1989 ax-9 1996 ax-10 2016 ax-11 2031 ax-12 2044 ax-13 2245 ax-ext 2601 ax-rep 4731 ax-sep 4741 ax-nul 4749 ax-pow 4803 ax-pr 4867 ax-un 6902 ax-cnex 9936 ax-resscn 9937 ax-1cn 9938 ax-icn 9939 ax-addcl 9940 ax-addrcl 9941 ax-mulcl 9942 ax-mulrcl 9943 ax-mulcom 9944 ax-addass 9945 ax-mulass 9946 ax-distr 9947 ax-i2m1 9948 ax-1ne0 9949 ax-1rid 9950 ax-rnegex 9951 ax-rrecex 9952 ax-cnre 9953 ax-pre-lttri 9954 ax-pre-lttrn 9955 ax-pre-ltadd 9956 ax-pre-mulgt0 9957

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-ex 1702 df-nf 1707 df-sb 1878 df-eu 2473 df-mo 2474 df-clab 2608 df-cleq 2614 df-clel 2617 df-nfc 2750 df-ne 2791 df-nel 2894 df-ral 2912 df-rex 2913 df-reu 2914 df-rab 2916 df-v 3188 df-sbc 3418 df-csb 3515 df-dif 3558 df-un 3560 df-in 3562 df-ss 3569 df-pss 3571 df-nul 3892 df-if 4059 df-pw 4132 df-sn 4149 df-pr 4151 df-tp 4153 df-op 4155 df-uni 4403 df-iun 4487 df-br 4614 df-opab 4674 df-mpt 4675 df-tr 4713 df-eprel 4985 df-id 4989 df-po 4995 df-so 4996 df-fr 5033 df-we 5035 df-xp 5080 df-rel 5081 df-cnv 5082 df-co 5083 df-dm 5084 df-rn 5085 df-res 5086 df-ima 5087 df-pred 5639 df-ord 5685 df-on 5686 df-lim 5687 df-suc 5688 df-iota 5810 df-fun 5849 df-fn 5850 df-f 5851 df-f1 5852 df-fo 5853 df-f1o 5854 df-fv 5855 df-riota 6565 df-ov 6607 df-oprab 6608 df-mpt2 6609 df-of 6850 df-om 7013 df-1st 7113 df-2nd 7114 df-wrecs 7352 df-recs 7413 df-rdg 7451 df-er 7687 df-map 7804 df-en 7900 df-dom 7901 df-sdom 7902 df-pnf 10020 df-mnf 10021 df-xr 10022 df-ltxr 10023 df-le 10024 df-sub 10212 df-neg 10213 df-nn 10965 df-n0 11237 df-z 11322 df-uz 11632 df-fz 12269 df-fzo 12407

This theorem is referenced by: poimirlem4 33045

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