Theorem canthwelem 10603

Description: Lemma for canthwe 10604. (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 2729	. . . . . . . 8 ⊢ 𝐵 = 𝐵
2		eqid 2729	. . . . . . . 8 ⊢ (𝑊‘𝐵) = (𝑊‘𝐵)
3	1, 2	pm3.2i 470	. . . . . . 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 482	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐴 ∈ 𝑉)
6		df-ov 7390	. . . . . . . . 9 ⊢ (𝑥𝐹𝑟) = (𝐹‘⟨𝑥, 𝑟⟩)
7		f1f 6756	. . . . . . . . . . 11 ⊢ (𝐹:𝑂–1-1→𝐴 → 𝐹:𝑂⟶𝐴)
8	7	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → 𝐹:𝑂⟶𝐴)
9		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
10		opabidw 5484	. . . . . . . . . . . 12 ⊢ (⟨𝑥, 𝑟⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
11	9, 10	sylibr 234	. . . . . . . . . . 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)})
12		canthwe.1	. . . . . . . . . . 11 ⊢ 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}
13	11, 12	eleqtrrdi 2839	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → ⟨𝑥, 𝑟⟩ ∈ 𝑂)
14	8, 13	ffvelcdmd 7057	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹‘⟨𝑥, 𝑟⟩) ∈ 𝐴)
15	6, 14	eqeltrid 2832	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
16		canthwe.3	. . . . . . . 8 ⊢ 𝐵 = ∪ dom 𝑊
17	4, 5, 15, 16	fpwwe2 10596	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝐵𝑊(𝑊‘𝐵) ∧ (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))))
18	3, 17	mpbiri 258	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵))
19	18	simprd 495	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵)
20		canthwe.4	. . . . . . . . . 10 ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})
21	20, 20	xpeq12i 5666	. . . . . . . . . . 11 ⊢ (𝐶 × 𝐶) = ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}))
22	21	ineq2i 4180	. . . . . . . . . 10 ⊢ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) = ((𝑊‘𝐵) ∩ ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})))
23	20, 22	oveq12i 7399	. . . . . . . . 9 ⊢ (𝐶𝐹((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) = ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})𝐹((𝑊‘𝐵) ∩ ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}))))
24	18	simpld 494	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐵𝑊(𝑊‘𝐵))
25	4, 5, 24	fpwwe2lem3 10586	. . . . . . . . . 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)))
26	19, 25	mpdan 687	. . . . . . . . 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)))
27	23, 26	eqtrid 2776	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐶𝐹((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) = (𝐵𝐹(𝑊‘𝐵)))
28		df-ov 7390	. . . . . . . 8 ⊢ (𝐶𝐹((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) = (𝐹‘⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩)
29		df-ov 7390	. . . . . . . 8 ⊢ (𝐵𝐹(𝑊‘𝐵)) = (𝐹‘⟨𝐵, (𝑊‘𝐵)⟩)
30	27, 28, 29	3eqtr3g 2787	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐹‘⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊‘𝐵)⟩))
31		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐹:𝑂–1-1→𝐴)
32		cnvimass 6053	. . . . . . . . . . . . 13 ⊢ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ⊆ dom (𝑊‘𝐵)
33	4, 5	fpwwe2lem2 10585	. . . . . . . . . . . . . . . . . 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))))
34	24, 33	mpbid 232	. . . . . . . . . . . . . . . . 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)))
35	34	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)))
36	35	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) ⊆ (𝐵 × 𝐵))
37		dmss 5866	. . . . . . . . . . . . . . 15 ⊢ ((𝑊‘𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
38	36, 37	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
39		dmxpss 6144	. . . . . . . . . . . . . 14 ⊢ dom (𝐵 × 𝐵) ⊆ 𝐵
40	38, 39	sstrdi 3959	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → dom (𝑊‘𝐵) ⊆ 𝐵)
41	32, 40	sstrid 3958	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ⊆ 𝐵)
42	20, 41	eqsstrid 3985	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐶 ⊆ 𝐵)
43	35	simpld 494	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐵 ⊆ 𝐴)
44	42, 43	sstrd 3957	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐶 ⊆ 𝐴)
45		inss2 4201	. . . . . . . . . . 11 ⊢ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)
46	45	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶))
47	34	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 [(◡(𝑊‘𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝐵) ∩ (𝑢 × 𝑢))) = 𝑦))
48	47	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) We 𝐵)
49		wess 5624	. . . . . . . . . . . 12 ⊢ (𝐶 ⊆ 𝐵 → ((𝑊‘𝐵) We 𝐵 → (𝑊‘𝐵) We 𝐶))
50	42, 48, 49	sylc 65	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) We 𝐶)
51		weinxp 5723	. . . . . . . . . . 11 ⊢ ((𝑊‘𝐵) We 𝐶 ↔ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
52	50, 51	sylib 218	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
53		fvex 6871	. . . . . . . . . . . . . 14 ⊢ (𝑊‘𝐵) ∈ V
54	53	cnvex 7901	. . . . . . . . . . . . 13 ⊢ ◡(𝑊‘𝐵) ∈ V
55	54	imaex 7890	. . . . . . . . . . . 12 ⊢ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ∈ V
56	20, 55	eqeltri 2824	. . . . . . . . . . 11 ⊢ 𝐶 ∈ V
57	53	inex1 5272	. . . . . . . . . . 11 ⊢ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ∈ V
58		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → 𝑥 = 𝐶)
59	58	sseq1d 3978	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑥 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐴))
60		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)))
61	58	sqxpeqd 5670	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑥 × 𝑥) = (𝐶 × 𝐶))
62	60, 61	sseq12d 3980	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)))
63	60, 58	weeq12d 5627	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 We 𝑥 ↔ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
64	59, 62, 63	3anbi123d 1438	. . . . . . . . . . 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)))
65	56, 57, 64	opelopaba 5496	. . . . . . . . . 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))
66	44, 46, 52, 65	syl3anbrc 1344	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
67	66, 12	eleqtrrdi 2839	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ 𝑂)
68	5, 43	ssexd 5279	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐵 ∈ V)
69	53	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) ∈ V)
70		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → 𝑥 = 𝐵)
71	70	sseq1d 3978	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴))
72		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → 𝑟 = (𝑊‘𝐵))
73	70	sqxpeqd 5670	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑥 × 𝑥) = (𝐵 × 𝐵))
74	72, 73	sseq12d 3980	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)))
75	72, 70	weeq12d 5627	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑟 We 𝑥 ↔ (𝑊‘𝐵) We 𝐵))
76	71, 74, 75	3anbi123d 1438	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊‘𝐵) We 𝐵)))
77	76	opelopabga 5493	. . . . . . . . . . 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)))
78	68, 69, 77	syl2anc 584	. . . . . . . . . 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)))
79	43, 36, 48, 78	mpbir3and 1343	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐵, (𝑊‘𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
80	79, 12	eleqtrrdi 2839	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐵, (𝑊‘𝐵)⟩ ∈ 𝑂)
81		f1fveq 7237	. . . . . . . 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)⟩))
82	31, 67, 80, 81	syl12anc 836	. . . . . . 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)⟩))
83	30, 82	mpbid 232	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊‘𝐵)⟩)
84	56, 57	opth1 5435	. . . . . 6 ⊢ (⟨𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊‘𝐵)⟩ → 𝐶 = 𝐵)
85	83, 84	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐶 = 𝐵)
86	19, 85	eleqtrrd 2831	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐶)
87	86, 20	eleqtrdi 2838	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵)) ∈ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}))
88		ovex 7420	. . . . 5 ⊢ (𝐵𝐹(𝑊‘𝐵)) ∈ V
89	88	eliniseg 6065	. . . 4 ⊢ ((𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵 → ((𝐵𝐹(𝑊‘𝐵)) ∈ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ↔ (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵))))
90	19, 89	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))))
91	87, 90	mpbid 232	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵)))
92		weso 5629	. . . 4 ⊢ ((𝑊‘𝐵) We 𝐵 → (𝑊‘𝐵) Or 𝐵)
93	48, 92	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝑊‘𝐵) Or 𝐵)
94		sonr 5570	. . 3 ⊢ (((𝑊‘𝐵) Or 𝐵 ∧ (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐵) → ¬ (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵)))
95	93, 19, 94	syl2anc 584	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → ¬ (𝐵𝐹(𝑊‘𝐵))(𝑊‘𝐵)(𝐵𝐹(𝑊‘𝐵)))
96	91, 95	pm2.65da 816	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:𝑂–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447  [wsbc 3753   ∩ cin 3913   ⊆ wss 3914  {csn 4589  ⟨cop 4595  ∪ cuni 4871   class class class wbr 5107  {copab 5169   Or wor 5545   We wwe 5590   × cxp 5636  ^◡ccnv 5637  dom cdm 5638   “ cima 5641  ⟶wf 6507  –1-1→wf1 6508  ‘cfv 6511  (class class class)co 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-oi 9463

This theorem is referenced by:  canthwe  10604