Proof of Theorem canthwelem
| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . . . . . 8
⊢ 𝐵 = 𝐵 |
| 2 | | eqid 2737 |
. . . . . . . 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 7434 |
. . . . . . . . 9
⊢ (𝑥𝐹𝑟) = (𝐹‘〈𝑥, 𝑟〉) |
| 7 | | f1f 6804 |
. . . . . . . . . . 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 5529 |
. . . . . . . . . . . 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 2852 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → 〈𝑥, 𝑟〉 ∈ 𝑂) |
| 14 | 8, 13 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝐹‘〈𝑥, 𝑟〉) ∈ 𝐴) |
| 15 | 6, 14 | eqeltrid 2845 |
. . . . . . . 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 10683 |
. . . . . . 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 5713 |
. . . . . . . . . . 11
⊢ (𝐶 × 𝐶) = ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})) |
| 22 | 21 | ineq2i 4217 |
. . . . . . . . . 10
⊢ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) = ((𝑊‘𝐵) ∩ ((◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) × (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}))) |
| 23 | 20, 22 | oveq12i 7443 |
. . . . . . . . 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 10673 |
. . . . . . . . . 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 2789 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐶𝐹((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) = (𝐵𝐹(𝑊‘𝐵))) |
| 28 | | df-ov 7434 |
. . . . . . . 8
⊢ (𝐶𝐹((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) = (𝐹‘〈𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))〉) |
| 29 | | df-ov 7434 |
. . . . . . . 8
⊢ (𝐵𝐹(𝑊‘𝐵)) = (𝐹‘〈𝐵, (𝑊‘𝐵)〉) |
| 30 | 27, 28, 29 | 3eqtr3g 2800 |
. . . . . . 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 6100 |
. . . . . . . . . . . . 13
⊢ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ⊆ dom (𝑊‘𝐵) |
| 33 | 4, 5 | fpwwe2lem2 10672 |
. . . . . . . . . . . . . . . . . 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 5913 |
. . . . . . . . . . . . . . 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 6191 |
. . . . . . . . . . . . . 14
⊢ dom
(𝐵 × 𝐵) ⊆ 𝐵 |
| 40 | 38, 39 | sstrdi 3996 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → dom (𝑊‘𝐵) ⊆ 𝐵) |
| 41 | 32, 40 | sstrid 3995 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ⊆ 𝐵) |
| 42 | 20, 41 | eqsstrid 4022 |
. . . . . . . . . . 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 3994 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 𝐶 ⊆ 𝐴) |
| 45 | | inss2 4238 |
. . . . . . . . . . 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 5671 |
. . . . . . . . . . . 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 5770 |
. . . . . . . . . . 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 6919 |
. . . . . . . . . . . . . 14
⊢ (𝑊‘𝐵) ∈ V |
| 54 | 53 | cnvex 7947 |
. . . . . . . . . . . . 13
⊢ ◡(𝑊‘𝐵) ∈ V |
| 55 | 54 | imaex 7936 |
. . . . . . . . . . . 12
⊢ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))}) ∈ V |
| 56 | 20, 55 | eqeltri 2837 |
. . . . . . . . . . 11
⊢ 𝐶 ∈ V |
| 57 | 53 | inex1 5317 |
. . . . . . . . . . 11
⊢ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ∈ V |
| 58 | | simpl 482 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → 𝑥 = 𝐶) |
| 59 | 58 | sseq1d 4015 |
. . . . . . . . . . . 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 5717 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑥 × 𝑥) = (𝐶 × 𝐶)) |
| 62 | 60, 61 | sseq12d 4017 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝐶 ∧ 𝑟 = ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ((𝑊‘𝐵) ∩ (𝐶 × 𝐶)) ⊆ (𝐶 × 𝐶))) |
| 63 | 60, 58 | weeq12d 5674 |
. . . . . . . . . . . 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 5541 |
. . . . . . . . . 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 2852 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 〈𝐶, ((𝑊‘𝐵) ∩ (𝐶 × 𝐶))〉 ∈ 𝑂) |
| 68 | 5, 43 | ssexd 5324 |
. . . . . . . . . . 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 4015 |
. . . . . . . . . . . . 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 5717 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑥 × 𝑥) = (𝐵 × 𝐵)) |
| 74 | 72, 73 | sseq12d 4017 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = 𝐵 ∧ 𝑟 = (𝑊‘𝐵)) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵))) |
| 75 | 72, 70 | weeq12d 5674 |
. . . . . . . . . . . . 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 5538 |
. . . . . . . . . . 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 2852 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → 〈𝐵, (𝑊‘𝐵)〉 ∈ 𝑂) |
| 81 | | f1fveq 7282 |
. . . . . . . 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 837 |
. . . . . . 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 5480 |
. . . . . 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 2844 |
. . . 4
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵)) ∈ 𝐶) |
| 87 | 86, 20 | eleqtrdi 2851 |
. . 3
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝑂–1-1→𝐴) → (𝐵𝐹(𝑊‘𝐵)) ∈ (◡(𝑊‘𝐵) “ {(𝐵𝐹(𝑊‘𝐵))})) |
| 88 | | ovex 7464 |
. . . . 5
⊢ (𝐵𝐹(𝑊‘𝐵)) ∈ V |
| 89 | 88 | eliniseg 6112 |
. . . 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 5676 |
. . . 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 5616 |
. . 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 817 |
1
⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:𝑂–1-1→𝐴) |