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Mirrors > Home > MPE Home > Th. List > inviso1 | Structured version Visualization version GIF version |
Description: If 𝐺 is an inverse to 𝐹, then 𝐹 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 |
---|---|
inviso1 | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
2 | invfval.n | . . . . 5 ⊢ 𝑁 = (Inv‘𝐶) | |
3 | invfval.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | invfval.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | invfval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | invfun 16809 | . . . 4 ⊢ (𝜑 → Fun (𝑋𝑁𝑌)) |
7 | funrel 6152 | . . . 4 ⊢ (Fun (𝑋𝑁𝑌) → Rel (𝑋𝑁𝑌)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → Rel (𝑋𝑁𝑌)) |
9 | inviso1.1 | . . 3 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)𝐺) | |
10 | releldm 5604 | . . 3 ⊢ ((Rel (𝑋𝑁𝑌) ∧ 𝐹(𝑋𝑁𝑌)𝐺) → 𝐹 ∈ dom (𝑋𝑁𝑌)) | |
11 | 8, 9, 10 | syl2anc 579 | . 2 ⊢ (𝜑 → 𝐹 ∈ dom (𝑋𝑁𝑌)) |
12 | isoval.n | . . 3 ⊢ 𝐼 = (Iso‘𝐶) | |
13 | 1, 2, 3, 4, 5, 12 | isoval 16810 | . 2 ⊢ (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌)) |
14 | 11, 13 | eleqtrrd 2861 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 class class class wbr 4886 dom cdm 5355 Rel wrel 5360 Fun wfun 6129 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 Catccat 16710 Invcinv 16790 Isociso 16791 |
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-rmo 3097 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-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-cat 16714 df-cid 16715 df-sect 16792 df-inv 16793 df-iso 16794 |
This theorem is referenced by: inviso2 16812 isoco 16822 idiso 16833 funciso 16919 ffthiso 16974 fuciso 17020 initoeu1 17046 termoeu1 17053 catciso 17142 yoneda 17309 |
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