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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 16803 | . 2 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)))) |
7 | simprl 761 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋) | |
8 | simprr 763 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌) | |
9 | 7, 8 | oveq12d 6940 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝑆𝑦) = (𝑋𝑆𝑌)) |
10 | 8, 7 | oveq12d 6940 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑦𝑆𝑥) = (𝑌𝑆𝑋)) |
11 | 10 | cnveqd 5543 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ◡(𝑦𝑆𝑥) = ◡(𝑌𝑆𝑋)) |
12 | 9, 11 | ineq12d 4037 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ((𝑥𝑆𝑦) ∩ ◡(𝑦𝑆𝑥)) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))) |
13 | invfval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
14 | ovex 6954 | . . . 4 ⊢ (𝑋𝑆𝑌) ∈ V | |
15 | 14 | inex1 5036 | . . 3 ⊢ ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)) ∈ V |
16 | 15 | a1i 11 | . 2 ⊢ (𝜑 → ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋)) ∈ V) |
17 | 6, 12, 4, 13, 16 | ovmpt2d 7065 | 1 ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ∩ cin 3790 ◡ccnv 5354 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 Catccat 16710 Sectcsect 16789 Invcinv 16790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-inv 16793 |
This theorem is referenced by: isinv 16805 invss 16806 dfiso2 16817 oppcinv 16825 |
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