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Mirrors > Home > MPE Home > Th. List > inviso2 | Structured version Visualization version GIF version |
Description: If 𝐺 is an inverse to 𝐹, then 𝐺 is an isomorphism. (Contributed by Mario Carneiro, 3-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) |
isoval.n | ⊢ 𝐼 = (Iso‘𝐶) |
inviso1.1 | ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) |
Ref | Expression |
---|---|
inviso2 | ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . 2 ⊢ 𝐵 = (Base‘𝐶) | |
2 | invfval.n | . 2 ⊢ 𝑁 = (Inv‘𝐶) | |
3 | invfval.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | invfval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
5 | invfval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | isoval.n | . 2 ⊢ 𝐼 = (Iso‘𝐶) | |
7 | inviso1.1 | . . 3 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) | |
8 | 1, 2, 3, 5, 4 | invsym 17810 | . . 3 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐺(𝑌𝑁𝑋)𝐹)) |
9 | 7, 8 | mpbid 232 | . 2 ⊢ (𝜑 → 𝐺(𝑌𝑁𝑋)𝐹) |
10 | 1, 2, 3, 4, 5, 6, 9 | inviso1 17814 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Catccat 17709 Invcinv 17793 Isociso 17794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-cat 17713 df-cid 17714 df-sect 17795 df-inv 17796 df-iso 17797 |
This theorem is referenced by: yonffthlem 18339 |
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