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| Mirrors > Home > MPE Home > Th. List > invf1o | Structured version Visualization version GIF version | ||
| Description: The inverse relation is a bijection from isomorphisms to isomorphisms. This means that every isomorphism 𝐹 ∈ (𝑋𝐼𝑌) has a unique inverse, denoted by ((Inv‘𝐶)‘𝐹). Remark 3.12 of [Adamek] p. 28. (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) | 
| isoval.n | ⊢ 𝐼 = (Iso‘𝐶) | 
| Ref | Expression | 
|---|---|
| invf1o | ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | invfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
| 2 | invfval.n | . . . 4 ⊢ 𝑁 = (Inv‘𝐶) | |
| 3 | invfval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 4 | invfval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 5 | invfval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 6 | isoval.n | . . . 4 ⊢ 𝐼 = (Iso‘𝐶) | |
| 7 | 1, 2, 3, 4, 5, 6 | invf 17813 | . . 3 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) | 
| 8 | 7 | ffnd 6736 | . 2 ⊢ (𝜑 → (𝑋𝑁𝑌) Fn (𝑋𝐼𝑌)) | 
| 9 | 1, 2, 3, 5, 4, 6 | invf 17813 | . . . 4 ⊢ (𝜑 → (𝑌𝑁𝑋):(𝑌𝐼𝑋)⟶(𝑋𝐼𝑌)) | 
| 10 | 9 | ffnd 6736 | . . 3 ⊢ (𝜑 → (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋)) | 
| 11 | 1, 2, 3, 4, 5 | invsym2 17808 | . . . 4 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) = (𝑌𝑁𝑋)) | 
| 12 | 11 | fneq1d 6660 | . . 3 ⊢ (𝜑 → (◡(𝑋𝑁𝑌) Fn (𝑌𝐼𝑋) ↔ (𝑌𝑁𝑋) Fn (𝑌𝐼𝑋))) | 
| 13 | 10, 12 | mpbird 257 | . 2 ⊢ (𝜑 → ◡(𝑋𝑁𝑌) Fn (𝑌𝐼𝑋)) | 
| 14 | dff1o4 6855 | . 2 ⊢ ((𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋) ↔ ((𝑋𝑁𝑌) Fn (𝑋𝐼𝑌) ∧ ◡(𝑋𝑁𝑌) Fn (𝑌𝐼𝑋))) | |
| 15 | 8, 13, 14 | sylanbrc 583 | 1 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)–1-1-onto→(𝑌𝐼𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ◡ccnv 5683 Fn wfn 6555 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Catccat 17708 Invcinv 17790 Isociso 17791 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-cat 17712 df-cid 17713 df-sect 17792 df-inv 17793 df-iso 17794 | 
| This theorem is referenced by: invinv 17815 | 
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