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| Mirrors > Home > MPE Home > Th. List > invfval | Structured version Visualization version GIF version | ||
| Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| invfval.b | ⊢ 𝐵 = (Base‘𝐶) | 
| invfval.n | ⊢ 𝑁 = (Inv‘𝐶) | 
| invfval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) | 
| invfval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| invfval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) | 
| invfval.s | ⊢ 𝑆 = (Sect‘𝐶) | 
| Ref | Expression | 
|---|---|
| invfval | ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | invfval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
| 2 | invfval.n | . . 3 ⊢ 𝑁 = (Inv‘𝐶) | |
| 3 | invfval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 4 | invfval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 5 | invfval.s | . . 3 ⊢ 𝑆 = (Sect‘𝐶) | |
| 6 | 1, 2, 3, 4, 4, 5 | invffval 17803 | . 2 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))) | 
| 7 | simprl 770 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋) | |
| 8 | simprr 772 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌) | |
| 9 | 7, 8 | oveq12d 7450 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝑆𝑦) = (𝑋𝑆𝑌)) | 
| 10 | 8, 7 | oveq12d 7450 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋)) | 
| 11 | 10 | cnveqd 5885 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ◡(𝑦𝑆𝑥) = ◡(𝑌𝑆𝑋)) | 
| 12 | 9, 11 | ineq12d 4220 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))) | 
| 13 | invfval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 14 | ovex 7465 | . . . 4 ⊢ (𝑋𝑆𝑌) ∈ V | |
| 15 | 14 | inex1 5316 | . . 3 ⊢ ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)) ∈ V | 
| 16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)) ∈ V) | 
| 17 | 6, 12, 4, 13, 16 | ovmpod 7586 | 1 ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 ◡ccnv 5683 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Catccat 17708 Sectcsect 17789 Invcinv 17790 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-inv 17793 | 
| This theorem is referenced by: isinv 17805 invss 17806 dfiso2 17817 oppcinv 17825 | 
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