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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 17150 | . . 3 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐺(𝑌𝑁𝑋)𝐹)) |
9 | 7, 8 | mpbid 235 | . 2 ⊢ (𝜑 → 𝐺(𝑌𝑁𝑋)𝐹) |
10 | 1, 2, 3, 4, 5, 6, 9 | inviso1 17154 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 class class class wbr 5040 ‘cfv 6350 (class class class)co 7183 Basecbs 16599 Catccat 17051 Invcinv 17133 Isociso 17134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-1st 7727 df-2nd 7728 df-cat 17055 df-cid 17056 df-sect 17135 df-inv 17136 df-iso 17137 |
This theorem is referenced by: yonffthlem 17661 |
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