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Theorem canthwelem 9510
Description: Lemma for canthwe 9511. (Contributed by Mario Carneiro, 31-May-2015.)
Hypotheses
Ref Expression
canthwe.1 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}
canthwe.2 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
canthwe.3 𝐵 = dom 𝑊
canthwe.4 𝐶 = ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})
Assertion
Ref Expression
canthwelem (𝐴𝑉 → ¬ 𝐹:𝑂1-1𝐴)
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
StepHypRef Expression
1 eqid 2651 . . . . . . . 8 𝐵 = 𝐵
2 eqid 2651 . . . . . . . 8 (𝑊𝐵) = (𝑊𝐵)
31, 2pm3.2i 470 . . . . . . 7 (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))
4 canthwe.2 . . . . . . . 8 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
5 elex 3243 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ V)
65adantr 480 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐴 ∈ V)
7 df-ov 6693 . . . . . . . . 9 (𝑥𝐹𝑟) = (𝐹‘⟨𝑥, 𝑟⟩)
8 f1f 6139 . . . . . . . . . . 11 (𝐹:𝑂1-1𝐴𝐹:𝑂𝐴)
98ad2antlr 763 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → 𝐹:𝑂𝐴)
10 simpr 476 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
11 opabid 5011 . . . . . . . . . . . 12 (⟨𝑥, 𝑟⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
1210, 11sylibr 224 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → ⟨𝑥, 𝑟⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
13 canthwe.1 . . . . . . . . . . 11 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}
1412, 13syl6eleqr 2741 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → ⟨𝑥, 𝑟⟩ ∈ 𝑂)
159, 14ffvelrnd 6400 . . . . . . . . 9 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹‘⟨𝑥, 𝑟⟩) ∈ 𝐴)
167, 15syl5eqel 2734 . . . . . . . 8 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
17 canthwe.3 . . . . . . . 8 𝐵 = dom 𝑊
184, 6, 16, 17fpwwe2 9503 . . . . . . 7 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
193, 18mpbiri 248 . . . . . 6 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝑊(𝑊𝐵) ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵))
2019simprd 478 . . . . 5 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ 𝐵)
21 canthwe.4 . . . . . . . . . 10 𝐶 = ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})
2221, 21xpeq12i 5171 . . . . . . . . . . 11 (𝐶 × 𝐶) = (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))
2322ineq2i 3844 . . . . . . . . . 10 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) = ((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})))
2421, 23oveq12i 6702 . . . . . . . . 9 (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})𝐹((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))))
2519simpld 474 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵𝑊(𝑊𝐵))
264, 6, 25fpwwe2lem3 9493 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵) → (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})𝐹((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})))) = (𝐵𝐹(𝑊𝐵)))
2720, 26mpdan 703 . . . . . . . . 9 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})𝐹((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})))) = (𝐵𝐹(𝑊𝐵)))
2824, 27syl5eq 2697 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (𝐵𝐹(𝑊𝐵)))
29 df-ov 6693 . . . . . . . 8 (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩)
30 df-ov 6693 . . . . . . . 8 (𝐵𝐹(𝑊𝐵)) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩)
3128, 29, 303eqtr3g 2708 . . . . . . 7 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩))
32 simpr 476 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐹:𝑂1-1𝐴)
33 cnvimass 5520 . . . . . . . . . . . . 13 ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ⊆ dom (𝑊𝐵)
344, 6fpwwe2lem2 9492 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 [((𝑊𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝐵) ∩ (𝑢 × 𝑢))) = 𝑦))))
3525, 34mpbid 222 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 [((𝑊𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝐵) ∩ (𝑢 × 𝑢))) = 𝑦)))
3635simpld 474 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
3736simprd 478 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) ⊆ (𝐵 × 𝐵))
38 dmss 5355 . . . . . . . . . . . . . . 15 ((𝑊𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
3937, 38syl 17 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐹:𝑂1-1𝐴) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
40 dmxpss 5600 . . . . . . . . . . . . . 14 dom (𝐵 × 𝐵) ⊆ 𝐵
4139, 40syl6ss 3648 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:𝑂1-1𝐴) → dom (𝑊𝐵) ⊆ 𝐵)
4233, 41syl5ss 3647 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ⊆ 𝐵)
4321, 42syl5eqss 3682 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶𝐵)
4436simpld 474 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵𝐴)
4543, 44sstrd 3646 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶𝐴)
46 inss2 3867 . . . . . . . . . . 11 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)
4746a1i 11 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶))
4835simprd 478 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 [((𝑊𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝐵) ∩ (𝑢 × 𝑢))) = 𝑦))
4948simpld 474 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) We 𝐵)
50 wess 5130 . . . . . . . . . . . 12 (𝐶𝐵 → ((𝑊𝐵) We 𝐵 → (𝑊𝐵) We 𝐶))
5143, 49, 50sylc 65 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) We 𝐶)
52 weinxp 5220 . . . . . . . . . . 11 ((𝑊𝐵) We 𝐶 ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
5351, 52sylib 208 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
54 fvex 6239 . . . . . . . . . . . . . 14 (𝑊𝐵) ∈ V
5554cnvex 7155 . . . . . . . . . . . . 13 (𝑊𝐵) ∈ V
5655imaex 7146 . . . . . . . . . . . 12 ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ∈ V
5721, 56eqeltri 2726 . . . . . . . . . . 11 𝐶 ∈ V
5854inex1 4832 . . . . . . . . . . 11 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ∈ V
59 simpl 472 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → 𝑥 = 𝐶)
6059sseq1d 3665 . . . . . . . . . . . 12 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑥𝐴𝐶𝐴))
61 simpr 476 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → 𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶)))
6259sqxpeqd 5175 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑥 × 𝑥) = (𝐶 × 𝐶))
6361, 62sseq12d 3667 . . . . . . . . . . . 12 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)))
64 weeq2 5132 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → (𝑟 We 𝑥𝑟 We 𝐶))
65 weeq1 5131 . . . . . . . . . . . . 13 (𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶)) → (𝑟 We 𝐶 ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
6664, 65sylan9bb 736 . . . . . . . . . . . 12 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 We 𝑥 ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
6760, 63, 663anbi123d 1439 . . . . . . . . . . 11 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝐶𝐴 ∧ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶) ∧ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)))
6857, 58, 67opelopaba 5020 . . . . . . . . . 10 (⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝐶𝐴 ∧ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶) ∧ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
6945, 47, 53, 68syl3anbrc 1265 . . . . . . . . 9 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
7069, 13syl6eleqr 2741 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ 𝑂)
716, 44ssexd 4838 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵 ∈ V)
7254a1i 11 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) ∈ V)
73 simpl 472 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → 𝑥 = 𝐵)
7473sseq1d 3665 . . . . . . . . . . . . 13 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑥𝐴𝐵𝐴))
75 simpr 476 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → 𝑟 = (𝑊𝐵))
7673sqxpeqd 5175 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑥 × 𝑥) = (𝐵 × 𝐵))
7775, 76sseq12d 3667 . . . . . . . . . . . . 13 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
78 weeq2 5132 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝑟 We 𝑥𝑟 We 𝐵))
79 weeq1 5131 . . . . . . . . . . . . . 14 (𝑟 = (𝑊𝐵) → (𝑟 We 𝐵 ↔ (𝑊𝐵) We 𝐵))
8078, 79sylan9bb 736 . . . . . . . . . . . . 13 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑟 We 𝑥 ↔ (𝑊𝐵) We 𝐵))
8174, 77, 803anbi123d 1439 . . . . . . . . . . . 12 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊𝐵) We 𝐵)))
8281opelopabga 5017 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ (𝑊𝐵) ∈ V) → (⟨𝐵, (𝑊𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊𝐵) We 𝐵)))
8371, 72, 82syl2anc 694 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (⟨𝐵, (𝑊𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊𝐵) We 𝐵)))
8444, 37, 49, 83mpbir3and 1264 . . . . . . . . 9 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐵, (𝑊𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
8584, 13syl6eleqr 2741 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐵, (𝑊𝐵)⟩ ∈ 𝑂)
86 f1fveq 6559 . . . . . . . 8 ((𝐹:𝑂1-1𝐴 ∧ (⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ 𝑂 ∧ ⟨𝐵, (𝑊𝐵)⟩ ∈ 𝑂)) → ((𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩) ↔ ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊𝐵)⟩))
8732, 70, 85, 86syl12anc 1364 . . . . . . 7 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩) ↔ ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊𝐵)⟩))
8831, 87mpbid 222 . . . . . 6 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊𝐵)⟩)
8957, 58opth1 4973 . . . . . 6 (⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊𝐵)⟩ → 𝐶 = 𝐵)
9088, 89syl 17 . . . . 5 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶 = 𝐵)
9120, 90eleqtrrd 2733 . . . 4 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ 𝐶)
9291, 21syl6eleq 2740 . . 3 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))
93 ovex 6718 . . . . 5 (𝐵𝐹(𝑊𝐵)) ∈ V
9493eliniseg 5529 . . . 4 ((𝐵𝐹(𝑊𝐵)) ∈ 𝐵 → ((𝐵𝐹(𝑊𝐵)) ∈ ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ↔ (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵))))
9520, 94syl 17 . . 3 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝐵𝐹(𝑊𝐵)) ∈ ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ↔ (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵))))
9692, 95mpbid 222 . 2 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵)))
97 weso 5134 . . . 4 ((𝑊𝐵) We 𝐵 → (𝑊𝐵) Or 𝐵)
9849, 97syl 17 . . 3 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) Or 𝐵)
99 sonr 5085 . . 3 (((𝑊𝐵) Or 𝐵 ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵) → ¬ (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵)))
10098, 20, 99syl2anc 694 . 2 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ¬ (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵)))
10196, 100pm2.65da 599 1 (𝐴𝑉 → ¬ 𝐹:𝑂1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  [wsbc 3468  cin 3606  wss 3607  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685  {copab 4745   Or wor 5063   We wwe 5101   × cxp 5141  ccnv 5142  dom cdm 5143  cima 5146  wf 5922  1-1wf1 5923  cfv 5926  (class class class)co 6690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-wrecs 7452  df-recs 7513  df-oi 8456
This theorem is referenced by:  canthwe  9511
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