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Theorem canthwelem 10066
Description: Lemma for canthwe 10067. (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 2826 . . . . . . . 8 𝐵 = 𝐵
2 eqid 2826 . . . . . . . 8 (𝑊𝐵) = (𝑊𝐵)
31, 2pm3.2i 471 . . . . . . 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 3518 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ V)
65adantr 481 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐴 ∈ V)
7 df-ov 7153 . . . . . . . . 9 (𝑥𝐹𝑟) = (𝐹‘⟨𝑥, 𝑟⟩)
8 f1f 6574 . . . . . . . . . . 11 (𝐹:𝑂1-1𝐴𝐹:𝑂𝐴)
98ad2antlr 723 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → 𝐹:𝑂𝐴)
10 simpr 485 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
11 opabidw 5409 . . . . . . . . . . . 12 (⟨𝑥, 𝑟⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
1210, 11sylibr 235 . . . . . . . . . . 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 2929 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → ⟨𝑥, 𝑟⟩ ∈ 𝑂)
159, 14ffvelrnd 6850 . . . . . . . . 9 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹‘⟨𝑥, 𝑟⟩) ∈ 𝐴)
167, 15eqeltrid 2922 . . . . . . . 8 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
17 canthwe.3 . . . . . . . 8 𝐵 = dom 𝑊
184, 6, 16, 17fpwwe2 10059 . . . . . . 7 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
193, 18mpbiri 259 . . . . . 6 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝑊(𝑊𝐵) ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵))
2019simprd 496 . . . . 5 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ 𝐵)
21 canthwe.4 . . . . . . . . . 10 𝐶 = ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})
2221, 21xpeq12i 5582 . . . . . . . . . . 11 (𝐶 × 𝐶) = (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))
2322ineq2i 4190 . . . . . . . . . 10 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) = ((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})))
2421, 23oveq12i 7162 . . . . . . . . 9 (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})𝐹((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))))
2519simpld 495 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵𝑊(𝑊𝐵))
264, 6, 25fpwwe2lem3 10049 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵) → (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})𝐹((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})))) = (𝐵𝐹(𝑊𝐵)))
2720, 26mpdan 683 . . . . . . . . 9 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})𝐹((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})))) = (𝐵𝐹(𝑊𝐵)))
2824, 27syl5eq 2873 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (𝐵𝐹(𝑊𝐵)))
29 df-ov 7153 . . . . . . . 8 (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩)
30 df-ov 7153 . . . . . . . 8 (𝐵𝐹(𝑊𝐵)) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩)
3128, 29, 303eqtr3g 2884 . . . . . . 7 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩))
32 simpr 485 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐹:𝑂1-1𝐴)
33 cnvimass 5948 . . . . . . . . . . . . 13 ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ⊆ dom (𝑊𝐵)
344, 6fpwwe2lem2 10048 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 [((𝑊𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝐵) ∩ (𝑢 × 𝑢))) = 𝑦))))
3525, 34mpbid 233 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 [((𝑊𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝐵) ∩ (𝑢 × 𝑢))) = 𝑦)))
3635simpld 495 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
3736simprd 496 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) ⊆ (𝐵 × 𝐵))
38 dmss 5770 . . . . . . . . . . . . . . 15 ((𝑊𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
3937, 38syl 17 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐹:𝑂1-1𝐴) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
40 dmxpss 6027 . . . . . . . . . . . . . 14 dom (𝐵 × 𝐵) ⊆ 𝐵
4139, 40sstrdi 3983 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:𝑂1-1𝐴) → dom (𝑊𝐵) ⊆ 𝐵)
4233, 41sstrid 3982 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ⊆ 𝐵)
4321, 42eqsstrid 4019 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶𝐵)
4436simpld 495 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵𝐴)
4543, 44sstrd 3981 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶𝐴)
46 inss2 4210 . . . . . . . . . . 11 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)
4746a1i 11 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶))
4835simprd 496 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 [((𝑊𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝐵) ∩ (𝑢 × 𝑢))) = 𝑦))
4948simpld 495 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) We 𝐵)
50 wess 5541 . . . . . . . . . . . 12 (𝐶𝐵 → ((𝑊𝐵) We 𝐵 → (𝑊𝐵) We 𝐶))
5143, 49, 50sylc 65 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) We 𝐶)
52 weinxp 5635 . . . . . . . . . . 11 ((𝑊𝐵) We 𝐶 ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
5351, 52sylib 219 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
54 fvex 6682 . . . . . . . . . . . . . 14 (𝑊𝐵) ∈ V
5554cnvex 7623 . . . . . . . . . . . . 13 (𝑊𝐵) ∈ V
5655imaex 7614 . . . . . . . . . . . 12 ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ∈ V
5721, 56eqeltri 2914 . . . . . . . . . . 11 𝐶 ∈ V
5854inex1 5218 . . . . . . . . . . 11 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ∈ V
59 simpl 483 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → 𝑥 = 𝐶)
6059sseq1d 4002 . . . . . . . . . . . 12 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑥𝐴𝐶𝐴))
61 simpr 485 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → 𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶)))
6259sqxpeqd 5586 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑥 × 𝑥) = (𝐶 × 𝐶))
6361, 62sseq12d 4004 . . . . . . . . . . . 12 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)))
64 weeq2 5543 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → (𝑟 We 𝑥𝑟 We 𝐶))
65 weeq1 5542 . . . . . . . . . . . . 13 (𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶)) → (𝑟 We 𝐶 ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
6664, 65sylan9bb 510 . . . . . . . . . . . 12 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 We 𝑥 ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
6760, 63, 663anbi123d 1429 . . . . . . . . . . 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 5420 . . . . . . . . . 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 1337 . . . . . . . . 9 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
7069, 13syl6eleqr 2929 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ 𝑂)
716, 44ssexd 5225 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵 ∈ V)
7254a1i 11 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) ∈ V)
73 simpl 483 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → 𝑥 = 𝐵)
7473sseq1d 4002 . . . . . . . . . . . . 13 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑥𝐴𝐵𝐴))
75 simpr 485 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → 𝑟 = (𝑊𝐵))
7673sqxpeqd 5586 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑥 × 𝑥) = (𝐵 × 𝐵))
7775, 76sseq12d 4004 . . . . . . . . . . . . 13 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
78 weeq2 5543 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝑟 We 𝑥𝑟 We 𝐵))
79 weeq1 5542 . . . . . . . . . . . . . 14 (𝑟 = (𝑊𝐵) → (𝑟 We 𝐵 ↔ (𝑊𝐵) We 𝐵))
8078, 79sylan9bb 510 . . . . . . . . . . . . 13 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑟 We 𝑥 ↔ (𝑊𝐵) We 𝐵))
8174, 77, 803anbi123d 1429 . . . . . . . . . . . 12 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊𝐵) We 𝐵)))
8281opelopabga 5417 . . . . . . . . . . 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 584 . . . . . . . . . 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 1336 . . . . . . . . 9 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐵, (𝑊𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
8584, 13syl6eleqr 2929 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐵, (𝑊𝐵)⟩ ∈ 𝑂)
86 f1fveq 7016 . . . . . . . 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 834 . . . . . . 7 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩) ↔ ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊𝐵)⟩))
8831, 87mpbid 233 . . . . . 6 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊𝐵)⟩)
8957, 58opth1 5364 . . . . . 6 (⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊𝐵)⟩ → 𝐶 = 𝐵)
9088, 89syl 17 . . . . 5 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶 = 𝐵)
9120, 90eleqtrrd 2921 . . . 4 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ 𝐶)
9291, 21syl6eleq 2928 . . 3 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))
93 ovex 7183 . . . . 5 (𝐵𝐹(𝑊𝐵)) ∈ V
9493eliniseg 5957 . . . 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 233 . 2 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵)))
97 weso 5545 . . . 4 ((𝑊𝐵) We 𝐵 → (𝑊𝐵) Or 𝐵)
9849, 97syl 17 . . 3 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) Or 𝐵)
99 sonr 5495 . . 3 (((𝑊𝐵) Or 𝐵 ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵) → ¬ (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵)))
10098, 20, 99syl2anc 584 . 2 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ¬ (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵)))
10196, 100pm2.65da 813 1 (𝐴𝑉 → ¬ 𝐹:𝑂1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  Vcvv 3500  [wsbc 3776  cin 3939  wss 3940  {csn 4564  cop 4570   cuni 4837   class class class wbr 5063  {copab 5125   Or wor 5472   We wwe 5512   × cxp 5552  ccnv 5553  dom cdm 5554  cima 5557  wf 6350  1-1wf1 6351  cfv 6354  (class class class)co 7150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-wrecs 7943  df-recs 8004  df-oi 8968
This theorem is referenced by:  canthwe  10067
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