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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 7285 and fcofo 7286. Formerly part of proof of fcof1o 7294. (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 7285 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴–1-1→𝐵) | |
4 | 1, 2, 3 | syl2anc 585 | . 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 7286 | . . 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 6551 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴–onto→𝐵)) | |
10 | 4, 8, 9 | sylanbrc 584 | 1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 I cid 5574 ↾ cres 5679 ∘ ccom 5681 ⟶wf 6540 –1-1→wf1 6541 –onto→wfo 6542 –1-1-onto→wf1o 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 |
This theorem is referenced by: 2fcoidinvd 7293 fcof1o 7294 2fvidf1od 7296 catciso 18061 pmtrff1o 19331 evpmodpmf1o 21149 |
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