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| Mirrors > Home > MPE Home > Th. List > fcof1od | Structured version Visualization version GIF version | ||
| Description: A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 7308 and fcofo 7309. Formerly part of proof of fcof1o 7317. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.) | 
| Ref | Expression | 
|---|---|
| fcof1od.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | 
| fcof1od.g | ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | 
| fcof1od.a | ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) | 
| fcof1od.b | ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) | 
| Ref | Expression | 
|---|---|
| fcof1od | ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fcof1od.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 2 | fcof1od.a | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) | |
| 3 | fcof1 7308 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴–1-1→𝐵) | |
| 4 | 1, 2, 3 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵) | 
| 5 | fcof1od.g | . . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | |
| 6 | fcof1od.b | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) | |
| 7 | fcofo 7309 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) → 𝐹:𝐴–onto→𝐵) | |
| 8 | 1, 5, 6, 7 | syl3anc 1372 | . 2 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐵) | 
| 9 | df-f1o 6567 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | |
| 10 | 4, 8, 9 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 I cid 5576 ↾ cres 5686 ∘ ccom 5688 ⟶wf 6556 –1-1→wf1 6557 –onto→wfo 6558 –1-1-onto→wf1o 6559 | 
| 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-sep 5295 ax-nul 5305 ax-pr 5431 | 
| 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-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 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 | 
| This theorem is referenced by: 2fcoidinvd 7316 fcof1o 7317 2fvidf1od 7319 catciso 18157 pmtrff1o 19482 evpmodpmf1o 21615 | 
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