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Theorem canthwelem 10507
Description: Lemma for canthwe 10508. (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 2736 . . . . . . . 8 𝐵 = 𝐵
2 eqid 2736 . . . . . . . 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 simpl 483 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐴𝑉)
6 df-ov 7340 . . . . . . . . 9 (𝑥𝐹𝑟) = (𝐹‘⟨𝑥, 𝑟⟩)
7 f1f 6721 . . . . . . . . . . 11 (𝐹:𝑂1-1𝐴𝐹:𝑂𝐴)
87ad2antlr 724 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → 𝐹:𝑂𝐴)
9 simpr 485 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
10 opabidw 5468 . . . . . . . . . . . 12 (⟨𝑥, 𝑟⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
119, 10sylibr 233 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → ⟨𝑥, 𝑟⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
12 canthwe.1 . . . . . . . . . . 11 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}
1311, 12eleqtrrdi 2848 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → ⟨𝑥, 𝑟⟩ ∈ 𝑂)
148, 13ffvelcdmd 7018 . . . . . . . . 9 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹‘⟨𝑥, 𝑟⟩) ∈ 𝐴)
156, 14eqeltrid 2841 . . . . . . . 8 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
16 canthwe.3 . . . . . . . 8 𝐵 = dom 𝑊
174, 5, 15, 16fpwwe2 10500 . . . . . . 7 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
183, 17mpbiri 257 . . . . . 6 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝑊(𝑊𝐵) ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵))
1918simprd 496 . . . . 5 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ 𝐵)
20 canthwe.4 . . . . . . . . . 10 𝐶 = ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})
2120, 20xpeq12i 5648 . . . . . . . . . . 11 (𝐶 × 𝐶) = (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))
2221ineq2i 4156 . . . . . . . . . 10 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) = ((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})))
2320, 22oveq12i 7349 . . . . . . . . 9 (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})𝐹((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))))
2418simpld 495 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵𝑊(𝑊𝐵))
254, 5, 24fpwwe2lem3 10490 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵) → (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})𝐹((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})))) = (𝐵𝐹(𝑊𝐵)))
2619, 25mpdan 684 . . . . . . . . 9 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})𝐹((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})))) = (𝐵𝐹(𝑊𝐵)))
2723, 26eqtrid 2788 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (𝐵𝐹(𝑊𝐵)))
28 df-ov 7340 . . . . . . . 8 (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩)
29 df-ov 7340 . . . . . . . 8 (𝐵𝐹(𝑊𝐵)) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩)
3027, 28, 293eqtr3g 2799 . . . . . . 7 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩))
31 simpr 485 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐹:𝑂1-1𝐴)
32 cnvimass 6019 . . . . . . . . . . . . 13 ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ⊆ dom (𝑊𝐵)
334, 5fpwwe2lem2 10489 . . . . . . . . . . . . . . . . . 18 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 [((𝑊𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝐵) ∩ (𝑢 × 𝑢))) = 𝑦))))
3424, 33mpbid 231 . . . . . . . . . . . . . . . . 17 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 [((𝑊𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝐵) ∩ (𝑢 × 𝑢))) = 𝑦)))
3534simpld 495 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
3635simprd 496 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) ⊆ (𝐵 × 𝐵))
37 dmss 5844 . . . . . . . . . . . . . . 15 ((𝑊𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
3836, 37syl 17 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐹:𝑂1-1𝐴) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
39 dmxpss 6109 . . . . . . . . . . . . . 14 dom (𝐵 × 𝐵) ⊆ 𝐵
4038, 39sstrdi 3944 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:𝑂1-1𝐴) → dom (𝑊𝐵) ⊆ 𝐵)
4132, 40sstrid 3943 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ⊆ 𝐵)
4220, 41eqsstrid 3980 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶𝐵)
4335simpld 495 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵𝐴)
4442, 43sstrd 3942 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶𝐴)
45 inss2 4176 . . . . . . . . . . 11 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)
4645a1i 11 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶))
4734simprd 496 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 [((𝑊𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝐵) ∩ (𝑢 × 𝑢))) = 𝑦))
4847simpld 495 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) We 𝐵)
49 wess 5607 . . . . . . . . . . . 12 (𝐶𝐵 → ((𝑊𝐵) We 𝐵 → (𝑊𝐵) We 𝐶))
5042, 48, 49sylc 65 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) We 𝐶)
51 weinxp 5702 . . . . . . . . . . 11 ((𝑊𝐵) We 𝐶 ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
5250, 51sylib 217 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
53 fvex 6838 . . . . . . . . . . . . . 14 (𝑊𝐵) ∈ V
5453cnvex 7840 . . . . . . . . . . . . 13 (𝑊𝐵) ∈ V
5554imaex 7831 . . . . . . . . . . . 12 ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ∈ V
5620, 55eqeltri 2833 . . . . . . . . . . 11 𝐶 ∈ V
5753inex1 5261 . . . . . . . . . . 11 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ∈ V
58 simpl 483 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → 𝑥 = 𝐶)
5958sseq1d 3963 . . . . . . . . . . . 12 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑥𝐴𝐶𝐴))
60 simpr 485 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → 𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶)))
6158sqxpeqd 5652 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑥 × 𝑥) = (𝐶 × 𝐶))
6260, 61sseq12d 3965 . . . . . . . . . . . 12 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)))
63 weeq2 5609 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → (𝑟 We 𝑥𝑟 We 𝐶))
64 weeq1 5608 . . . . . . . . . . . . 13 (𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶)) → (𝑟 We 𝐶 ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
6563, 64sylan9bb 510 . . . . . . . . . . . 12 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 We 𝑥 ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
6659, 62, 653anbi123d 1435 . . . . . . . . . . 11 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝐶𝐴 ∧ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶) ∧ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)))
6756, 57, 66opelopaba 5480 . . . . . . . . . 10 (⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝐶𝐴 ∧ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶) ∧ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
6844, 46, 52, 67syl3anbrc 1342 . . . . . . . . 9 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
6968, 12eleqtrrdi 2848 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ 𝑂)
705, 43ssexd 5268 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵 ∈ V)
7153a1i 11 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) ∈ V)
72 simpl 483 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → 𝑥 = 𝐵)
7372sseq1d 3963 . . . . . . . . . . . . 13 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑥𝐴𝐵𝐴))
74 simpr 485 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → 𝑟 = (𝑊𝐵))
7572sqxpeqd 5652 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑥 × 𝑥) = (𝐵 × 𝐵))
7674, 75sseq12d 3965 . . . . . . . . . . . . 13 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
77 weeq2 5609 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝑟 We 𝑥𝑟 We 𝐵))
78 weeq1 5608 . . . . . . . . . . . . . 14 (𝑟 = (𝑊𝐵) → (𝑟 We 𝐵 ↔ (𝑊𝐵) We 𝐵))
7977, 78sylan9bb 510 . . . . . . . . . . . . 13 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑟 We 𝑥 ↔ (𝑊𝐵) We 𝐵))
8073, 76, 793anbi123d 1435 . . . . . . . . . . . 12 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊𝐵) We 𝐵)))
8180opelopabga 5477 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ (𝑊𝐵) ∈ V) → (⟨𝐵, (𝑊𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊𝐵) We 𝐵)))
8270, 71, 81syl2anc 584 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (⟨𝐵, (𝑊𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵) ∧ (𝑊𝐵) We 𝐵)))
8343, 36, 48, 82mpbir3and 1341 . . . . . . . . 9 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐵, (𝑊𝐵)⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
8483, 12eleqtrrdi 2848 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐵, (𝑊𝐵)⟩ ∈ 𝑂)
85 f1fveq 7191 . . . . . . . 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)⟩))
8631, 69, 84, 85syl12anc 834 . . . . . . 7 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩) ↔ ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊𝐵)⟩))
8730, 86mpbid 231 . . . . . 6 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊𝐵)⟩)
8856, 57opth1 5420 . . . . . 6 (⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊𝐵)⟩ → 𝐶 = 𝐵)
8987, 88syl 17 . . . . 5 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶 = 𝐵)
9019, 89eleqtrrd 2840 . . . 4 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ 𝐶)
9190, 20eleqtrdi 2847 . . 3 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))
92 ovex 7370 . . . . 5 (𝐵𝐹(𝑊𝐵)) ∈ V
9392eliniseg 6032 . . . 4 ((𝐵𝐹(𝑊𝐵)) ∈ 𝐵 → ((𝐵𝐹(𝑊𝐵)) ∈ ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ↔ (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵))))
9419, 93syl 17 . . 3 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝐵𝐹(𝑊𝐵)) ∈ ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ↔ (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵))))
9591, 94mpbid 231 . 2 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵)))
96 weso 5611 . . . 4 ((𝑊𝐵) We 𝐵 → (𝑊𝐵) Or 𝐵)
9748, 96syl 17 . . 3 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) Or 𝐵)
98 sonr 5555 . . 3 (((𝑊𝐵) Or 𝐵 ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵) → ¬ (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵)))
9997, 19, 98syl2anc 584 . 2 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ¬ (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵)))
10095, 99pm2.65da 814 1 (𝐴𝑉 → ¬ 𝐹:𝑂1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wral 3061  Vcvv 3441  [wsbc 3727  cin 3897  wss 3898  {csn 4573  cop 4579   cuni 4852   class class class wbr 5092  {copab 5154   Or wor 5531   We wwe 5574   × cxp 5618  ccnv 5619  dom cdm 5620  cima 5623  wf 6475  1-1wf1 6476  cfv 6479  (class class class)co 7337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488  df-riota 7293  df-ov 7340  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-oi 9367
This theorem is referenced by:  canthwe  10508
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