| Step | Hyp | Ref
| Expression |
| 1 | | elrel 5808 |
. . . . . . . . . . 11
⊢ ((Rel
𝐹 ∧ 𝑝 ∈ 𝐹) → ∃𝑥∃𝑦 𝑝 = 〈𝑥, 𝑦〉) |
| 2 | 1 | ex 412 |
. . . . . . . . . 10
⊢ (Rel
𝐹 → (𝑝 ∈ 𝐹 → ∃𝑥∃𝑦 𝑝 = 〈𝑥, 𝑦〉)) |
| 3 | | elrel 5808 |
. . . . . . . . . . 11
⊢ ((Rel
𝐹 ∧ 𝑞 ∈ 𝐹) → ∃𝑎∃𝑧 𝑞 = 〈𝑎, 𝑧〉) |
| 4 | 3 | ex 412 |
. . . . . . . . . 10
⊢ (Rel
𝐹 → (𝑞 ∈ 𝐹 → ∃𝑎∃𝑧 𝑞 = 〈𝑎, 𝑧〉)) |
| 5 | 2, 4 | anim12d 609 |
. . . . . . . . 9
⊢ (Rel
𝐹 → ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) → (∃𝑥∃𝑦 𝑝 = 〈𝑥, 𝑦〉 ∧ ∃𝑎∃𝑧 𝑞 = 〈𝑎, 𝑧〉))) |
| 6 | 5 | adantrd 491 |
. . . . . . . 8
⊢ (Rel
𝐹 → (((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)
→ (∃𝑥∃𝑦 𝑝 = 〈𝑥, 𝑦〉 ∧ ∃𝑎∃𝑧 𝑞 = 〈𝑎, 𝑧〉))) |
| 7 | 6 | pm4.71rd 562 |
. . . . . . 7
⊢ (Rel
𝐹 → (((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)
↔ ((∃𝑥∃𝑦 𝑝 = 〈𝑥, 𝑦〉 ∧ ∃𝑎∃𝑧 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)))) |
| 8 | | 19.41vvvv 1952 |
. . . . . . . 8
⊢
(∃𝑥∃𝑦∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ (∃𝑥∃𝑦∃𝑎∃𝑧(𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 9 | | ee4anv 2353 |
. . . . . . . . 9
⊢
(∃𝑥∃𝑦∃𝑎∃𝑧(𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ↔ (∃𝑥∃𝑦 𝑝 = 〈𝑥, 𝑦〉 ∧ ∃𝑎∃𝑧 𝑞 = 〈𝑎, 𝑧〉)) |
| 10 | 9 | anbi1i 624 |
. . . . . . . 8
⊢
((∃𝑥∃𝑦∃𝑎∃𝑧(𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ((∃𝑥∃𝑦 𝑝 = 〈𝑥, 𝑦〉 ∧ ∃𝑎∃𝑧 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 11 | 8, 10 | bitr2i 276 |
. . . . . . 7
⊢
(((∃𝑥∃𝑦 𝑝 = 〈𝑥, 𝑦〉 ∧ ∃𝑎∃𝑧 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ∃𝑥∃𝑦∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 12 | 7, 11 | bitrdi 287 |
. . . . . 6
⊢ (Rel
𝐹 → (((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)
↔ ∃𝑥∃𝑦∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)))) |
| 13 | 12 | 2exbidv 1924 |
. . . . 5
⊢ (Rel
𝐹 → (∃𝑝∃𝑞((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)
↔ ∃𝑝∃𝑞∃𝑥∃𝑦∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)))) |
| 14 | | excom13 2164 |
. . . . . 6
⊢
(∃𝑝∃𝑞∃𝑥∃𝑦∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ∃𝑥∃𝑞∃𝑝∃𝑦∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 15 | | excom13 2164 |
. . . . . . . 8
⊢
(∃𝑞∃𝑝∃𝑦∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ∃𝑦∃𝑝∃𝑞∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 16 | | exrot4 2166 |
. . . . . . . . . 10
⊢
(∃𝑝∃𝑞∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ∃𝑎∃𝑧∃𝑝∃𝑞((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 17 | | excom 2162 |
. . . . . . . . . 10
⊢
(∃𝑎∃𝑧∃𝑝∃𝑞((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ∃𝑧∃𝑎∃𝑝∃𝑞((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 18 | | df-3an 1089 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉 ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 19 | 18 | 2exbii 1849 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑝∃𝑞(𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉 ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ∃𝑝∃𝑞((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 20 | | opex 5469 |
. . . . . . . . . . . . . . . 16
⊢
〈𝑥, 𝑦〉 ∈ V |
| 21 | | opex 5469 |
. . . . . . . . . . . . . . . 16
⊢
〈𝑎, 𝑧〉 ∈ V |
| 22 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝑝 ∈ 𝐹 ↔ 〈𝑥, 𝑦〉 ∈ 𝐹)) |
| 23 | 22 | anbi1d 631 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 = 〈𝑥, 𝑦〉 → ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ↔ (〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹))) |
| 24 | | breq2 5147 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 = 〈𝑥, 𝑦〉 → (𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝
↔ 𝑞(1st
⊗ ((V ∖ I ) ∘ 2nd ))〈𝑥, 𝑦〉)) |
| 25 | 23, 24 | anbi12d 632 |
. . . . . . . . . . . . . . . 16
⊢ (𝑝 = 〈𝑥, 𝑦〉 → (((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)
↔ ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))〈𝑥, 𝑦〉))) |
| 26 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑞 = 〈𝑎, 𝑧〉 → (𝑞 ∈ 𝐹 ↔ 〈𝑎, 𝑧〉 ∈ 𝐹)) |
| 27 | 26 | anbi2d 630 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑞 = 〈𝑎, 𝑧〉 → ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ↔ (〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑎, 𝑧〉 ∈ 𝐹))) |
| 28 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑞 = 〈𝑎, 𝑧〉 → (𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))〈𝑥, 𝑦〉 ↔ 〈𝑎, 𝑧〉(1st ⊗ ((V ∖ I
) ∘ 2nd ))〈𝑥, 𝑦〉)) |
| 29 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑥 ∈ V |
| 30 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑦 ∈ V |
| 31 | 21, 29, 30 | brtxp 35881 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(〈𝑎, 𝑧〉(1st ⊗
((V ∖ I ) ∘ 2nd ))〈𝑥, 𝑦〉 ↔ (〈𝑎, 𝑧〉1st 𝑥 ∧ 〈𝑎, 𝑧〉((V ∖ I ) ∘ 2nd
)𝑦)) |
| 32 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑎 ∈ V |
| 33 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑧 ∈ V |
| 34 | 32, 33 | br1steq 35771 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(〈𝑎, 𝑧〉1st 𝑥 ↔ 𝑥 = 𝑎) |
| 35 | | equcom 2017 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑎 ↔ 𝑎 = 𝑥) |
| 36 | 34, 35 | bitri 275 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(〈𝑎, 𝑧〉1st 𝑥 ↔ 𝑎 = 𝑥) |
| 37 | 21, 30 | brco 5881 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(〈𝑎, 𝑧〉((V ∖ I ) ∘
2nd )𝑦 ↔
∃𝑥(〈𝑎, 𝑧〉2nd 𝑥 ∧ 𝑥(V ∖ I )𝑦)) |
| 38 | 32, 33 | br2ndeq 35772 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(〈𝑎, 𝑧〉2nd 𝑥 ↔ 𝑥 = 𝑧) |
| 39 | 38 | anbi1i 624 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((〈𝑎, 𝑧〉2nd 𝑥 ∧ 𝑥(V ∖ I )𝑦) ↔ (𝑥 = 𝑧 ∧ 𝑥(V ∖ I )𝑦)) |
| 40 | 39 | exbii 1848 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑥(〈𝑎, 𝑧〉2nd 𝑥 ∧ 𝑥(V ∖ I )𝑦) ↔ ∃𝑥(𝑥 = 𝑧 ∧ 𝑥(V ∖ I )𝑦)) |
| 41 | | breq1 5146 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 𝑧 → (𝑥(V ∖ I )𝑦 ↔ 𝑧(V ∖ I )𝑦)) |
| 42 | | brv 5477 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 𝑧V𝑦 |
| 43 | | brdif 5196 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧(V ∖ I )𝑦 ↔ (𝑧V𝑦 ∧ ¬ 𝑧 I 𝑦)) |
| 44 | 42, 43 | mpbiran 709 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑧 I 𝑦) |
| 45 | 30 | ideq 5863 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑧 I 𝑦 ↔ 𝑧 = 𝑦) |
| 46 | | equcom 2017 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑧 = 𝑦 ↔ 𝑦 = 𝑧) |
| 47 | 45, 46 | bitri 275 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 I 𝑦 ↔ 𝑦 = 𝑧) |
| 48 | 47 | notbii 320 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (¬
𝑧 I 𝑦 ↔ ¬ 𝑦 = 𝑧) |
| 49 | 44, 48 | bitri 275 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧) |
| 50 | 41, 49 | bitrdi 287 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑧 → (𝑥(V ∖ I )𝑦 ↔ ¬ 𝑦 = 𝑧)) |
| 51 | 50 | equsexvw 2004 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑥(𝑥 = 𝑧 ∧ 𝑥(V ∖ I )𝑦) ↔ ¬ 𝑦 = 𝑧) |
| 52 | 37, 40, 51 | 3bitri 297 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(〈𝑎, 𝑧〉((V ∖ I ) ∘
2nd )𝑦 ↔
¬ 𝑦 = 𝑧) |
| 53 | 36, 52 | anbi12i 628 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((〈𝑎, 𝑧〉1st 𝑥 ∧ 〈𝑎, 𝑧〉((V ∖ I ) ∘ 2nd
)𝑦) ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧)) |
| 54 | 31, 53 | bitri 275 |
. . . . . . . . . . . . . . . . . . 19
⊢
(〈𝑎, 𝑧〉(1st ⊗
((V ∖ I ) ∘ 2nd ))〈𝑥, 𝑦〉 ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧)) |
| 55 | 28, 54 | bitrdi 287 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑞 = 〈𝑎, 𝑧〉 → (𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))〈𝑥, 𝑦〉 ↔ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧))) |
| 56 | 27, 55 | anbi12d 632 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑞 = 〈𝑎, 𝑧〉 → (((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))〈𝑥, 𝑦〉) ↔ ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑎, 𝑧〉 ∈ 𝐹) ∧ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧)))) |
| 57 | | an12 645 |
. . . . . . . . . . . . . . . . 17
⊢
(((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑎, 𝑧〉 ∈ 𝐹) ∧ (𝑎 = 𝑥 ∧ ¬ 𝑦 = 𝑧)) ↔ (𝑎 = 𝑥 ∧ ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑎, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))) |
| 58 | 56, 57 | bitrdi 287 |
. . . . . . . . . . . . . . . 16
⊢ (𝑞 = 〈𝑎, 𝑧〉 → (((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))〈𝑥, 𝑦〉) ↔ (𝑎 = 𝑥 ∧ ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑎, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)))) |
| 59 | 20, 21, 25, 58 | ceqsex2v 3536 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑝∃𝑞(𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉 ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ (𝑎 = 𝑥 ∧ ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑎, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))) |
| 60 | 19, 59 | bitr3i 277 |
. . . . . . . . . . . . . 14
⊢
(∃𝑝∃𝑞((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ (𝑎 = 𝑥 ∧ ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑎, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))) |
| 61 | 60 | exbii 1848 |
. . . . . . . . . . . . 13
⊢
(∃𝑎∃𝑝∃𝑞((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ∃𝑎(𝑎 = 𝑥 ∧ ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑎, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))) |
| 62 | | opeq1 4873 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑥 → 〈𝑎, 𝑧〉 = 〈𝑥, 𝑧〉) |
| 63 | 62 | eleq1d 2826 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑥 → (〈𝑎, 𝑧〉 ∈ 𝐹 ↔ 〈𝑥, 𝑧〉 ∈ 𝐹)) |
| 64 | 63 | anbi2d 630 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑥 → ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑎, 𝑧〉 ∈ 𝐹) ↔ (〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹))) |
| 65 | 64 | anbi1d 631 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑥 → (((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑎, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧) ↔ ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧))) |
| 66 | 65 | equsexvw 2004 |
. . . . . . . . . . . . 13
⊢
(∃𝑎(𝑎 = 𝑥 ∧ ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑎, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)) ↔ ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)) |
| 67 | 61, 66 | bitri 275 |
. . . . . . . . . . . 12
⊢
(∃𝑎∃𝑝∃𝑞((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)) |
| 68 | 67 | exbii 1848 |
. . . . . . . . . . 11
⊢
(∃𝑧∃𝑎∃𝑝∃𝑞((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ∃𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧)) |
| 69 | | exanali 1859 |
. . . . . . . . . . 11
⊢
(∃𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) ∧ ¬ 𝑦 = 𝑧) ↔ ¬ ∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) |
| 70 | 68, 69 | bitri 275 |
. . . . . . . . . 10
⊢
(∃𝑧∃𝑎∃𝑝∃𝑞((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ¬ ∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) |
| 71 | 16, 17, 70 | 3bitri 297 |
. . . . . . . . 9
⊢
(∃𝑝∃𝑞∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ¬ ∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) |
| 72 | 71 | exbii 1848 |
. . . . . . . 8
⊢
(∃𝑦∃𝑝∃𝑞∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ∃𝑦 ¬ ∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) |
| 73 | | exnal 1827 |
. . . . . . . 8
⊢
(∃𝑦 ¬
∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) |
| 74 | 15, 72, 73 | 3bitri 297 |
. . . . . . 7
⊢
(∃𝑞∃𝑝∃𝑦∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ¬ ∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) |
| 75 | 74 | exbii 1848 |
. . . . . 6
⊢
(∃𝑥∃𝑞∃𝑝∃𝑦∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ∃𝑥 ¬ ∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) |
| 76 | | exnal 1827 |
. . . . . 6
⊢
(∃𝑥 ¬
∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) |
| 77 | 14, 75, 76 | 3bitri 297 |
. . . . 5
⊢
(∃𝑝∃𝑞∃𝑥∃𝑦∃𝑎∃𝑧((𝑝 = 〈𝑥, 𝑦〉 ∧ 𝑞 = 〈𝑎, 𝑧〉) ∧ ((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) ↔ ¬ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) |
| 78 | 13, 77 | bitrdi 287 |
. . . 4
⊢ (Rel
𝐹 → (∃𝑝∃𝑞((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)
↔ ¬ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧))) |
| 79 | 78 | con2bid 354 |
. . 3
⊢ (Rel
𝐹 → (∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧) ↔ ¬ ∃𝑝∃𝑞((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 80 | 79 | pm5.32i 574 |
. 2
⊢ ((Rel
𝐹 ∧ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧)) ↔ (Rel 𝐹 ∧ ¬ ∃𝑝∃𝑞((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 81 | | dffun4 6577 |
. 2
⊢ (Fun
𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((〈𝑥, 𝑦〉 ∈ 𝐹 ∧ 〈𝑥, 𝑧〉 ∈ 𝐹) → 𝑦 = 𝑧))) |
| 82 | | df-funs 35862 |
. . . 4
⊢ Funs = (𝒫 (V × V) ∖ Fix ( E ∘ ((1st ⊗ ((V ∖ I
) ∘ 2nd )) ∘ ◡ E
))) |
| 83 | 82 | eleq2i 2833 |
. . 3
⊢ (𝐹 ∈
Funs ↔ 𝐹
∈ (𝒫 (V × V) ∖ Fix ( E
∘ ((1st ⊗ ((V ∖ I ) ∘ 2nd ))
∘ ◡ E )))) |
| 84 | | eldif 3961 |
. . 3
⊢ (𝐹 ∈ (𝒫 (V × V)
∖ Fix ( E ∘ ((1st ⊗
((V ∖ I ) ∘ 2nd )) ∘ ◡ E ))) ↔ (𝐹 ∈ 𝒫 (V × V) ∧ ¬
𝐹 ∈ Fix ( E ∘ ((1st ⊗ ((V ∖ I
) ∘ 2nd )) ∘ ◡ E
)))) |
| 85 | | elfuns.1 |
. . . . . 6
⊢ 𝐹 ∈ V |
| 86 | 85 | elpw 4604 |
. . . . 5
⊢ (𝐹 ∈ 𝒫 (V × V)
↔ 𝐹 ⊆ (V ×
V)) |
| 87 | | df-rel 5692 |
. . . . 5
⊢ (Rel
𝐹 ↔ 𝐹 ⊆ (V × V)) |
| 88 | 86, 87 | bitr4i 278 |
. . . 4
⊢ (𝐹 ∈ 𝒫 (V × V)
↔ Rel 𝐹) |
| 89 | 85 | elfix 35904 |
. . . . . 6
⊢ (𝐹 ∈
Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘
2nd )) ∘ ◡ E )) ↔
𝐹( E ∘
((1st ⊗ ((V ∖ I ) ∘ 2nd )) ∘
◡ E ))𝐹) |
| 90 | 85, 85 | coep 35752 |
. . . . . . 7
⊢ (𝐹( E ∘ ((1st
⊗ ((V ∖ I ) ∘ 2nd )) ∘ ◡ E ))𝐹 ↔ ∃𝑝 ∈ 𝐹 𝐹((1st ⊗ ((V ∖ I )
∘ 2nd )) ∘ ◡ E
)𝑝) |
| 91 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑝 ∈ V |
| 92 | 85, 91 | coepr 35753 |
. . . . . . . 8
⊢ (𝐹((1st ⊗ ((V
∖ I ) ∘ 2nd )) ∘ ◡ E )𝑝 ↔ ∃𝑞 ∈ 𝐹 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝) |
| 93 | 92 | rexbii 3094 |
. . . . . . 7
⊢
(∃𝑝 ∈
𝐹 𝐹((1st ⊗ ((V ∖ I )
∘ 2nd )) ∘ ◡ E
)𝑝 ↔ ∃𝑝 ∈ 𝐹 ∃𝑞 ∈ 𝐹 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝) |
| 94 | 90, 93 | bitri 275 |
. . . . . 6
⊢ (𝐹( E ∘ ((1st
⊗ ((V ∖ I ) ∘ 2nd )) ∘ ◡ E ))𝐹 ↔ ∃𝑝 ∈ 𝐹 ∃𝑞 ∈ 𝐹 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝) |
| 95 | | r2ex 3196 |
. . . . . 6
⊢
(∃𝑝 ∈
𝐹 ∃𝑞 ∈ 𝐹 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝
↔ ∃𝑝∃𝑞((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) |
| 96 | 89, 94, 95 | 3bitri 297 |
. . . . 5
⊢ (𝐹 ∈
Fix ( E ∘ ((1st ⊗ ((V ∖ I ) ∘
2nd )) ∘ ◡ E )) ↔
∃𝑝∃𝑞((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) |
| 97 | 96 | notbii 320 |
. . . 4
⊢ (¬
𝐹 ∈ Fix ( E ∘ ((1st ⊗ ((V ∖ I
) ∘ 2nd )) ∘ ◡ E
)) ↔ ¬ ∃𝑝∃𝑞((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝)) |
| 98 | 88, 97 | anbi12i 628 |
. . 3
⊢ ((𝐹 ∈ 𝒫 (V × V)
∧ ¬ 𝐹 ∈ Fix ( E ∘ ((1st ⊗ ((V ∖ I
) ∘ 2nd )) ∘ ◡ E
))) ↔ (Rel 𝐹 ∧
¬ ∃𝑝∃𝑞((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 99 | 83, 84, 98 | 3bitri 297 |
. 2
⊢ (𝐹 ∈
Funs ↔ (Rel 𝐹
∧ ¬ ∃𝑝∃𝑞((𝑝 ∈ 𝐹 ∧ 𝑞 ∈ 𝐹) ∧ 𝑞(1st ⊗ ((V ∖ I )
∘ 2nd ))𝑝))) |
| 100 | 80, 81, 99 | 3bitr4ri 304 |
1
⊢ (𝐹 ∈
Funs ↔ Fun 𝐹) |