Theorem canthwelem 10602

Description: Lemma for canthwe 10603. (Contributed by Mario Carneiro, 31-May-2015.)

Hypotheses

Ref	Expression
canthwe.1	⊢ 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}
canthwe.2	⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
canthwe.3	⊢ 𝐵 = ∪ dom 𝑊
canthwe.4	⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})

Assertion

Ref	Expression
canthwelem	⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:𝑂–1-1→𝐴)

Distinct variable groups:   𝑢,𝑟,𝑥,𝑦,𝐵   𝐶,𝑟,𝑥   𝑂,𝑟,𝑢,𝑥,𝑦   𝑉,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑢,𝑥,𝑦   𝐹,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑦,𝑢)

Proof of Theorem canthwelem

Step	Hyp	Ref	Expression
1		eqid 2761	. . . . . . . 8 ⊢ 𝐵 = 𝐵
2		eqid 2761	. . . . . . . 8 ⊢ (𝑊‘𝐵) = (𝑊‘𝐵)
3	1, 2	pm3.2i 474	. . . . . . 7 ⊢ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))
4		canthwe.2	. . . . . . . 8 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
5		simpl 486	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐴 ∈ 𝑉)
6		df-ov 7394	. . . . . . . . 9 ⊢ (𝑥𝐹𝑟) = (𝐹‘⟨𝑥, 𝑟⟩)
7		f1f 6755	. . . . . . . . . . 11 ⊢ (𝐹:𝑂–1-1→𝐴 → 𝐹:𝑂⟶𝐴)
8	7	ad2antlr 737	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → 𝐹:𝑂⟶𝐴)
9		opabidw 5491	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑟⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
10	9	bilanri 510	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → ⟨𝑥, 𝑟⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
11		canthwe.1	. . . . . . . . . . 11 ⊢ 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}
12	10, 11	eleqtrrdi 2872	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → ⟨𝑥, 𝑟⟩ ∈ 𝑂)
13	8, 12	ffvelcdmd 7061	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹‘⟨𝑥, 𝑟⟩) ∈ 𝐴)
14	6, 13	eqeltrid 2865	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
15		canthwe.3	. . . . . . . 8 ⊢ 𝐵 = ∪ dom 𝑊
16	4, 5, 14, 15	fpwwe2 10595	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝐵𝑊(𝑊‘𝐵) ∧ (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))))
17	3, 16	mpbiri 260	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵))
18	17	simprd 499	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵)
19		canthwe.4	. . . . . . . . . 10 ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})
20	19, 19	xpeq12i 5671	. . . . . . . . . . 11 ⊢ (𝐶 × 𝐶) = ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}))
21	20	ineq2i 4167	. . . . . . . . . 10 ⊢ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) = ((𝑊‘𝐵) ∩ ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})))
22	19, 21	oveq12i 7403	. . . . . . . . 9 ⊢ (𝐶𝐹((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) = ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})𝐹((𝑊‘𝐵) ∩ ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}))))
23	17	simpld 498	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐵𝑊(𝑊‘𝐵))
24	4, 5, 23	fpwwe2lem3 10585	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵) → ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})𝐹((𝑊‘𝐵) ∩ ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})))) = (𝐵𝐹(𝑊‘𝐵)))
25	18, 24	mpdan 697	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})𝐹((𝑊‘𝐵) ∩ ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})))) = (𝐵𝐹(𝑊‘𝐵)))
26	22, 25	eqtrid 2808	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐶𝐹((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) = (𝐵𝐹(𝑊‘𝐵)))
27		df-ov 7394	. . . . . . . 8 ⊢ (𝐶𝐹((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) = (𝐹‘⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩)
28		df-ov 7394	. . . . . . . 8 ⊢ (𝐵𝐹(𝑊‘𝐵)) = (𝐹‘⟨𝐵, (𝑊‘𝐵)⟩)
29	26, 27, 28	3eqtr3g 2819	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐹‘⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊‘𝐵)⟩))
30		simpr 488	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐹:𝑂–1-1→𝐴)
31		cnvimass 6067	. . . . . . . . . . . . 13 ⊢ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ⊆ dom (𝑊‘𝐵)
32	4, 5	fpwwe2lem2 10584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝑊(𝑊‘𝐵) ↔ ((𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 [(◡(𝑊‘𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝐵) ∩ (𝑢 × 𝑢))) = 𝑦))))
33	23, 32	mpbid 234	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 [(◡(𝑊‘𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝐵) ∩ (𝑢 × 𝑢))) = 𝑦)))
34	33	simpld 498	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)))
35	34	simprd 499	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) ⊆ (𝐵 × 𝐵))
36		dmss 5874	. . . . . . . . . . . . . . 15 ⊢ ((𝑊‘𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
37	35, 36	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
38		dmxpss 6152	. . . . . . . . . . . . . 14 ⊢ dom (𝐵 × 𝐵) ⊆ 𝐵
39	37, 38	sstrdi 3946	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → dom (𝑊‘𝐵) ⊆ 𝐵)
40	31, 39	sstrid 3945	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ⊆ 𝐵)
41	19, 40	eqsstrid 3972	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐶 ⊆ 𝐵)
42	34	simpld 498	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐵 ⊆ 𝐴)
43	41, 42	sstrd 3944	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐶 ⊆ 𝐴)
44		inss2 4187	. . . . . . . . . . 11 ⊢ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)
45	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶))
46	33	simprd 499	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 [(◡(𝑊‘𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝐵) ∩ (𝑢 × 𝑢))) = 𝑦))
47	46	simpld 498	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) We 𝐵)
48		wess 5629	. . . . . . . . . . . 12 ⊢ (𝐶 ⊆ 𝐵 → ((𝑊‘𝐵) We 𝐵 → (𝑊‘𝐵) We 𝐶))
49	41, 47, 48	sylc 65	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) We 𝐶)
50		weinxp 5728	. . . . . . . . . . 11 ⊢ ((𝑊‘𝐵) We 𝐶 ↔ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
51	49, 50	sylib 220	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
52		fvex 6875	. . . . . . . . . . . . . 14 ⊢ (𝑊‘𝐵) ∈ V
53	52	cnvex 7901	. . . . . . . . . . . . 13 ⊢ ◡(𝑊‘𝐵) ∈ V
54	53	imaex 7890	. . . . . . . . . . . 12 ⊢ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ∈ V
55	19, 54	eqeltri 2857	. . . . . . . . . . 11 ⊢ 𝐶 ∈ V
56	52	inex1 5270	. . . . . . . . . . 11 ⊢ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ∈ V
57		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → 𝑥 = 𝐶)
58	57	sseq1d 3965	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑥 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐴))
59		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)))
60	57	sqxpeqd 5675	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑥 × 𝑥) = (𝐶 × 𝐶))
61	59, 60	sseq12d 3967	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)))
62	59, 57	weeq12d 5632	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 We 𝑥 ↔ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
63	58, 61, 62	3anbi123d 1456	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝐶 ⊆ 𝐴 ∧ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶) ∧ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)))
64	55, 56, 63	opelopaba 5503	. . . . . . . . . 10 ⊢ (⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝐶 ⊆ 𝐴 ∧ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶) ∧ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
65	43, 45, 51, 64	syl3anbrc 1356	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
66	65, 11	eleqtrrdi 2872	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ 𝑂)
67	5, 42	ssexd 5277	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐵 ∈ V)
68	52	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) ∈ V)
69		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → 𝑥 = 𝐵)
70	69	sseq1d 3965	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
71		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → 𝑟 = (𝑊‘𝐵))
72	69	sqxpeqd 5675	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑥 × 𝑥) = (𝐵 × 𝐵))
73	71, 72	sseq12d 3967	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)))
74	71, 69	weeq12d 5632	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑟 We 𝑥 ↔ (𝑊‘𝐵) We 𝐵))
75	70, 73, 74	3anbi123d 1456	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊‘𝐵) We 𝐵)))
76	75	opelopabga 5500	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ V ∧ (𝑊‘𝐵) ∈ V) → (⟨𝐵, (𝑊‘𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊‘𝐵) We 𝐵)))
77	67, 68, 76	syl2anc 593	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (⟨𝐵, (𝑊‘𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊‘𝐵) We 𝐵)))
78	42, 35, 47, 77	mpbir3and 1355	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐵, (𝑊‘𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
79	78, 11	eleqtrrdi 2872	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐵, (𝑊‘𝐵)⟩ ∈ 𝑂)
80		f1fveq 7241	. . . . . . . 8 ⊢ ((𝐹:𝑂–1-1→𝐴 ∧ (⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ 𝑂 ∧ ⟨𝐵, (𝑊‘𝐵)⟩ ∈ 𝑂)) → ((𝐹‘⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊‘𝐵)⟩) ↔ ⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊‘𝐵)⟩))
81	30, 66, 79, 80	syl12anc 847	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝐹‘⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊‘𝐵)⟩) ↔ ⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊‘𝐵)⟩))
82	29, 81	mpbid 234	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊‘𝐵)⟩)
83	55, 56	opth1 5440	. . . . . 6 ⊢ (⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊‘𝐵)⟩ → 𝐶 = 𝐵)
84	82, 83	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐶 = 𝐵)
85	18, 84	eleqtrrd 2864	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐶)
86	85, 19	eleqtrdi 2871	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵)) ∈ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}))
87		ovex 7424	. . . . 5 ⊢ (𝐵𝐹(𝑊‘𝐵)) ∈ V
88	87	eliniseg 6079	. . . 4 ⊢ ((𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵 → ((𝐵𝐹(𝑊‘𝐵)) ∈ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ↔ (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵))))
89	18, 88	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝐵𝐹(𝑊‘𝐵)) ∈ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ↔ (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵))))
90	86, 89	mpbid 234	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵)))
91		weso 5634	. . . 4 ⊢ ((𝑊‘𝐵) We 𝐵 → (𝑊‘𝐵) Or 𝐵)
92	47, 91	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) Or 𝐵)
93		sonr 5575	. . 3 ⊢ (((𝑊‘𝐵) Or 𝐵 ∧ (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵) → ¬ (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵)))
94	92, 18, 93	syl2anc 593	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ¬ (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵)))
95	90, 94	pm2.65da 826	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:𝑂–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453  [wsbc 3742   ∩ cin 3901   ⊆ wss 3902  {csn 4579  ⟨cop 4585  ∪ cuni 4862   class class class wbr 5097  {copab 5159   Or wor 5550   We wwe 5595   × cxp 5641  ^◡ccnv 5642  dom cdm 5643   “ cima 5646  ⟶wf 6512  –1-1→wf1 6513  ‘cfv 6516  (class class class)co 7391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-se 5597  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-isom 6525  df-riota 7348  df-ov 7394  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-oi 9452

This theorem is referenced by:  canthwe  10603