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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 17035 | . . 3 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐺(𝑌𝑁𝑋)𝐹)) |
9 | 7, 8 | mpbid 234 | . 2 ⊢ (𝜑 → 𝐺(𝑌𝑁𝑋)𝐹) |
10 | 1, 2, 3, 4, 5, 6, 9 | inviso1 17039 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 Catccat 16938 Invcinv 17018 Isociso 17019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-cat 16942 df-cid 16943 df-sect 17020 df-inv 17021 df-iso 17022 |
This theorem is referenced by: yonffthlem 17535 |
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