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Theorem canthwelem 10648
Description: Lemma for canthwe 10649. (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 2731 . . . . . . . 8 𝐵 = 𝐵
2 eqid 2731 . . . . . . . 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 simpl 482 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐴𝑉)
6 df-ov 7415 . . . . . . . . 9 (𝑥𝐹𝑟) = (𝐹‘⟨𝑥, 𝑟⟩)
7 f1f 6788 . . . . . . . . . . 11 (𝐹:𝑂1-1𝐴𝐹:𝑂𝐴)
87ad2antlr 724 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → 𝐹:𝑂𝐴)
9 simpr 484 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥))
10 opabidw 5525 . . . . . . . . . . . 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 2843 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → ⟨𝑥, 𝑟⟩ ∈ 𝑂)
148, 13ffvelcdmd 7088 . . . . . . . . 9 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹‘⟨𝑥, 𝑟⟩) ∈ 𝐴)
156, 14eqeltrid 2836 . . . . . . . 8 (((𝐴𝑉𝐹:𝑂1-1𝐴) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
16 canthwe.3 . . . . . . . 8 𝐵 = dom 𝑊
174, 5, 15, 16fpwwe2 10641 . . . . . . 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 495 . . . . 5 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ 𝐵)
20 canthwe.4 . . . . . . . . . 10 𝐶 = ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})
2120, 20xpeq12i 5705 . . . . . . . . . . 11 (𝐶 × 𝐶) = (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))
2221ineq2i 4210 . . . . . . . . . 10 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) = ((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})))
2320, 22oveq12i 7424 . . . . . . . . 9 (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))})𝐹((𝑊𝐵) ∩ (((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) × ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))))
2418simpld 494 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵𝑊(𝑊𝐵))
254, 5, 24fpwwe2lem3 10631 . . . . . . . . . 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 2783 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (𝐵𝐹(𝑊𝐵)))
28 df-ov 7415 . . . . . . . 8 (𝐶𝐹((𝑊𝐵) ∩ (𝐶 × 𝐶))) = (𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩)
29 df-ov 7415 . . . . . . . 8 (𝐵𝐹(𝑊𝐵)) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩)
3027, 28, 293eqtr3g 2794 . . . . . . 7 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐹‘⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩) = (𝐹‘⟨𝐵, (𝑊𝐵)⟩))
31 simpr 484 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐹:𝑂1-1𝐴)
32 cnvimass 6081 . . . . . . . . . . . . 13 ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ⊆ dom (𝑊𝐵)
334, 5fpwwe2lem2 10630 . . . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . . 16 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
3635simprd 495 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) ⊆ (𝐵 × 𝐵))
37 dmss 5903 . . . . . . . . . . . . . . 15 ((𝑊𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
3836, 37syl 17 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐹:𝑂1-1𝐴) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
39 dmxpss 6171 . . . . . . . . . . . . . 14 dom (𝐵 × 𝐵) ⊆ 𝐵
4038, 39sstrdi 3995 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:𝑂1-1𝐴) → dom (𝑊𝐵) ⊆ 𝐵)
4132, 40sstrid 3994 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ⊆ 𝐵)
4220, 41eqsstrid 4031 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶𝐵)
4335simpld 494 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵𝐴)
4442, 43sstrd 3993 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶𝐴)
45 inss2 4230 . . . . . . . . . . 11 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)
4645a1i 11 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶))
4734simprd 495 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 [((𝑊𝐵) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊𝐵) ∩ (𝑢 × 𝑢))) = 𝑦))
4847simpld 494 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) We 𝐵)
49 wess 5664 . . . . . . . . . . . 12 (𝐶𝐵 → ((𝑊𝐵) We 𝐵 → (𝑊𝐵) We 𝐶))
5042, 48, 49sylc 65 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) We 𝐶)
51 weinxp 5761 . . . . . . . . . . 11 ((𝑊𝐵) We 𝐶 ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
5250, 51sylib 217 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶)
53 fvex 6905 . . . . . . . . . . . . . 14 (𝑊𝐵) ∈ V
5453cnvex 7919 . . . . . . . . . . . . 13 (𝑊𝐵) ∈ V
5554imaex 7910 . . . . . . . . . . . 12 ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}) ∈ V
5620, 55eqeltri 2828 . . . . . . . . . . 11 𝐶 ∈ V
5753inex1 5318 . . . . . . . . . . 11 ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ∈ V
58 simpl 482 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → 𝑥 = 𝐶)
5958sseq1d 4014 . . . . . . . . . . . 12 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑥𝐴𝐶𝐴))
60 simpr 484 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → 𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶)))
6158sqxpeqd 5709 . . . . . . . . . . . . 13 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑥 × 𝑥) = (𝐶 × 𝐶))
6260, 61sseq12d 4016 . . . . . . . . . . . 12 ((𝑥 = 𝐶𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶)))
63 weeq2 5666 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → (𝑟 We 𝑥𝑟 We 𝐶))
64 weeq1 5665 . . . . . . . . . . . . 13 (𝑟 = ((𝑊𝐵) ∩ (𝐶 × 𝐶)) → (𝑟 We 𝐶 ↔ ((𝑊𝐵) ∩ (𝐶 × 𝐶)) We 𝐶))
6563, 64sylan9bb 509 . . . . . . . . . . . 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 5537 . . . . . . . . . 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 2843 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ ∈ 𝑂)
705, 43ssexd 5325 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐵 ∈ V)
7153a1i 11 . . . . . . . . . . 11 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) ∈ V)
72 simpl 482 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → 𝑥 = 𝐵)
7372sseq1d 4014 . . . . . . . . . . . . 13 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑥𝐴𝐵𝐴))
74 simpr 484 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → 𝑟 = (𝑊𝐵))
7572sqxpeqd 5709 . . . . . . . . . . . . . 14 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑥 × 𝑥) = (𝐵 × 𝐵))
7674, 75sseq12d 4016 . . . . . . . . . . . . 13 ((𝑥 = 𝐵𝑟 = (𝑊𝐵)) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
77 weeq2 5666 . . . . . . . . . . . . . 14 (𝑥 = 𝐵 → (𝑟 We 𝑥𝑟 We 𝐵))
78 weeq1 5665 . . . . . . . . . . . . . 14 (𝑟 = (𝑊𝐵) → (𝑟 We 𝐵 ↔ (𝑊𝐵) We 𝐵))
7977, 78sylan9bb 509 . . . . . . . . . . . . 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 5534 . . . . . . . . . . 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 583 . . . . . . . . . 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 2843 . . . . . . . 8 ((𝐴𝑉𝐹:𝑂1-1𝐴) → ⟨𝐵, (𝑊𝐵)⟩ ∈ 𝑂)
85 f1fveq 7264 . . . . . . . 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 5476 . . . . . 6 (⟨𝐶, ((𝑊𝐵) ∩ (𝐶 × 𝐶))⟩ = ⟨𝐵, (𝑊𝐵)⟩ → 𝐶 = 𝐵)
8987, 88syl 17 . . . . 5 ((𝐴𝑉𝐹:𝑂1-1𝐴) → 𝐶 = 𝐵)
9019, 89eleqtrrd 2835 . . . 4 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ 𝐶)
9190, 20eleqtrdi 2842 . . 3 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝐵𝐹(𝑊𝐵)) ∈ ((𝑊𝐵) “ {(𝐵𝐹(𝑊𝐵))}))
92 ovex 7445 . . . . 5 (𝐵𝐹(𝑊𝐵)) ∈ V
9392eliniseg 6094 . . . 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 5668 . . . 4 ((𝑊𝐵) We 𝐵 → (𝑊𝐵) Or 𝐵)
9748, 96syl 17 . . 3 ((𝐴𝑉𝐹:𝑂1-1𝐴) → (𝑊𝐵) Or 𝐵)
98 sonr 5612 . . 3 (((𝑊𝐵) Or 𝐵 ∧ (𝐵𝐹(𝑊𝐵)) ∈ 𝐵) → ¬ (𝐵𝐹(𝑊𝐵))(𝑊𝐵)(𝐵𝐹(𝑊𝐵)))
9997, 19, 98syl2anc 583 . 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 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  Vcvv 3473  [wsbc 3778  cin 3948  wss 3949  {csn 4629  cop 4635   cuni 4909   class class class wbr 5149  {copab 5211   Or wor 5588   We wwe 5631   × cxp 5675  ccnv 5676  dom cdm 5677  cima 5680  wf 6540  1-1wf1 6541  cfv 6544  (class class class)co 7412
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 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7368  df-ov 7415  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-oi 9508
This theorem is referenced by:  canthwe  10649
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