Theorem canthwelem 9472

Description: Lemma for canthwe 9473. (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 2622	. . . . . . . 8 ⊢ 𝐵 = 𝐵
2		eqid 2622	. . . . . . . 8 ⊢ (𝑊‘𝐵) = (𝑊‘𝐵)
3	1, 2	pm3.2i 471	. . . . . . 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		elex 3212	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
6	5	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐴 ∈ V)
7		df-ov 6653	. . . . . . . . 9 ⊢ (𝑥𝐹𝑟) = (𝐹‘⟨𝑥, 𝑟⟩)
8		f1f 6101	. . . . . . . . . . 11 ⊢ (𝐹:𝑂–1-1→𝐴 → 𝐹:𝑂⟶𝐴)
9	8	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → 𝐹:𝑂⟶𝐴)
10		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
11		opabid 4982	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑟⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
12	10, 11	sylibr 224	. . . . . . . . . . 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)})
13		canthwe.1	. . . . . . . . . . 11 ⊢ 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}
14	12, 13	syl6eleqr 2712	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → ⟨𝑥, 𝑟⟩ ∈ 𝑂)
15	9, 14	ffvelrnd 6360	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹‘⟨𝑥, 𝑟⟩) ∈ 𝐴)
16	7, 15	syl5eqel 2705	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
17		canthwe.3	. . . . . . . 8 ⊢ 𝐵 = ∪ dom 𝑊
18	4, 6, 16, 17	fpwwe2 9465	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝐵𝑊(𝑊‘𝐵) ∧ (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))))
19	3, 18	mpbiri 248	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵))
20	19	simprd 479	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵)
21		canthwe.4	. . . . . . . . . 10 ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})
22	21, 21	xpeq12i 5137	. . . . . . . . . . 11 ⊢ (𝐶 × 𝐶) = ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}))
23	22	ineq2i 3811	. . . . . . . . . 10 ⊢ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) = ((𝑊‘𝐵) ∩ ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})))
24	21, 23	oveq12i 6662	. . . . . . . . 9 ⊢ (𝐶𝐹((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) = ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})𝐹((𝑊‘𝐵) ∩ ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}))))
25	19	simpld 475	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐵𝑊(𝑊‘𝐵))
26	4, 6, 25	fpwwe2lem3 9455	. . . . . . . . . 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)))
27	20, 26	mpdan 702	. . . . . . . . 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)))
28	24, 27	syl5eq 2668	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐶𝐹((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) = (𝐵𝐹(𝑊‘𝐵)))
29		df-ov 6653	. . . . . . . 8 ⊢ (𝐶𝐹((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) = (𝐹‘⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩)
30		df-ov 6653	. . . . . . . 8 ⊢ (𝐵𝐹(𝑊‘𝐵)) = (𝐹‘⟨𝐵, (𝑊‘𝐵)⟩)
31	28, 29, 30	3eqtr3g 2679	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐹‘⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊‘𝐵)⟩))
32		simpr 477	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐹:𝑂–1-1→𝐴)
33		cnvimass 5485	. . . . . . . . . . . . 13 ⊢ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ⊆ dom (𝑊‘𝐵)
34	4, 6	fpwwe2lem2 9454	. . . . . . . . . . . . . . . . . 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))))
35	25, 34	mpbid 222	. . . . . . . . . . . . . . . . 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)))
36	35	simpld 475	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)))
37	36	simprd 479	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) ⊆ (𝐵 × 𝐵))
38		dmss 5323	. . . . . . . . . . . . . . 15 ⊢ ((𝑊‘𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
39	37, 38	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
40		dmxpss 5565	. . . . . . . . . . . . . 14 ⊢ dom (𝐵 × 𝐵) ⊆ 𝐵
41	39, 40	syl6ss 3615	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → dom (𝑊‘𝐵) ⊆ 𝐵)
42	33, 41	syl5ss 3614	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ⊆ 𝐵)
43	21, 42	syl5eqss 3649	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐶 ⊆ 𝐵)
44	36	simpld 475	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐵 ⊆ 𝐴)
45	43, 44	sstrd 3613	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐶 ⊆ 𝐴)
46		inss2 3834	. . . . . . . . . . 11 ⊢ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)
47	46	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶))
48	35	simprd 479	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 [(◡(𝑊‘𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝐵) ∩ (𝑢 × 𝑢))) = 𝑦))
49	48	simpld 475	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) We 𝐵)
50		wess 5101	. . . . . . . . . . . 12 ⊢ (𝐶 ⊆ 𝐵 → ((𝑊‘𝐵) We 𝐵 → (𝑊‘𝐵) We 𝐶))
51	43, 49, 50	sylc 65	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) We 𝐶)
52		weinxp 5186	. . . . . . . . . . 11 ⊢ ((𝑊‘𝐵) We 𝐶 ↔ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
53	51, 52	sylib 208	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
54		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (𝑊‘𝐵) ∈ V
55	54	cnvex 7113	. . . . . . . . . . . . 13 ⊢ ◡(𝑊‘𝐵) ∈ V
56	55	imaex 7104	. . . . . . . . . . . 12 ⊢ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ∈ V
57	21, 56	eqeltri 2697	. . . . . . . . . . 11 ⊢ 𝐶 ∈ V
58	54	inex1 4799	. . . . . . . . . . 11 ⊢ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ∈ V
59		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → 𝑥 = 𝐶)
60	59	sseq1d 3632	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑥 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐴))
61		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)))
62	59	sqxpeqd 5141	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑥 × 𝑥) = (𝐶 × 𝐶))
63	61, 62	sseq12d 3634	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)))
64		weeq2 5103	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐶 → (𝑟 We 𝑥 ↔ 𝑟 We 𝐶))
65		weeq1 5102	. . . . . . . . . . . . 13 ⊢ (𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) → (𝑟 We 𝐶 ↔ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
66	64, 65	sylan9bb 736	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 We 𝑥 ↔ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
67	60, 63, 66	3anbi123d 1399	. . . . . . . . . . 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)))
68	57, 58, 67	opelopaba 4991	. . . . . . . . . 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))
69	45, 47, 53, 68	syl3anbrc 1246	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
70	69, 13	syl6eleqr 2712	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ 𝑂)
71	6, 44	ssexd 4805	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐵 ∈ V)
72	54	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) ∈ V)
73		simpl 473	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → 𝑥 = 𝐵)
74	73	sseq1d 3632	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
75		simpr 477	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → 𝑟 = (𝑊‘𝐵))
76	73	sqxpeqd 5141	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑥 × 𝑥) = (𝐵 × 𝐵))
77	75, 76	sseq12d 3634	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)))
78		weeq2 5103	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐵 → (𝑟 We 𝑥 ↔ 𝑟 We 𝐵))
79		weeq1 5102	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑊‘𝐵) → (𝑟 We 𝐵 ↔ (𝑊‘𝐵) We 𝐵))
80	78, 79	sylan9bb 736	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑟 We 𝑥 ↔ (𝑊‘𝐵) We 𝐵))
81	74, 77, 80	3anbi123d 1399	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊‘𝐵) We 𝐵)))
82	81	opelopabga 4988	. . . . . . . . . . 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)))
83	71, 72, 82	syl2anc 693	. . . . . . . . . 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)))
84	44, 37, 49, 83	mpbir3and 1245	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐵, (𝑊‘𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
85	84, 13	syl6eleqr 2712	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐵, (𝑊‘𝐵)⟩ ∈ 𝑂)
86		f1fveq 6519	. . . . . . . 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)⟩))
87	32, 70, 85, 86	syl12anc 1324	. . . . . . 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)⟩))
88	31, 87	mpbid 222	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊‘𝐵)⟩)
89	57, 58	opth1 4944	. . . . . 6 ⊢ (⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊‘𝐵)⟩ → 𝐶 = 𝐵)
90	88, 89	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐶 = 𝐵)
91	20, 90	eleqtrrd 2704	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐶)
92	91, 21	syl6eleq 2711	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵)) ∈ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}))
93		ovex 6678	. . . . 5 ⊢ (𝐵𝐹(𝑊‘𝐵)) ∈ V
94	93	eliniseg 5494	. . . 4 ⊢ ((𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵 → ((𝐵𝐹(𝑊‘𝐵)) ∈ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ↔ (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵))))
95	20, 94	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))))
96	92, 95	mpbid 222	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵)))
97		weso 5105	. . . 4 ⊢ ((𝑊‘𝐵) We 𝐵 → (𝑊‘𝐵) Or 𝐵)
98	49, 97	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) Or 𝐵)
99		sonr 5056	. . 3 ⊢ (((𝑊‘𝐵) Or 𝐵 ∧ (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵) → ¬ (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵)))
100	98, 20, 99	syl2anc 693	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ¬ (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵)))
101	96, 100	pm2.65da 600	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:𝑂–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  [wsbc 3435   ∩ cin 3573   ⊆ wss 3574  {csn 4177  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653  {copab 4712   Or wor 5034   We wwe 5072   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-wrecs 7407  df-recs 7468  df-oi 8415

This theorem is referenced by:  canthwe  9473