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Mirrors > Home > MPE Home > Th. List > 2fcoidinvd | Structured version Visualization version GIF version |
Description: Show that a function is the inverse of a function if their compositions are the identity functions. (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 |
---|---|
2fcoidinvd | ⊢ (𝜑 → ◡𝐹 = 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcof1od.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
2 | fcof1od.g | . . 3 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | |
3 | fcof1od.a | . . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) | |
4 | fcof1od.b | . . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) | |
5 | 1, 2, 3, 4 | fcof1od 7236 | . 2 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
6 | 5, 2, 4 | fcof1oinvd 7235 | 1 ⊢ (𝜑 → ◡𝐹 = 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 I cid 5528 ◡ccnv 5630 ↾ cres 5633 ∘ ccom 5635 ⟶wf 6489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 |
This theorem is referenced by: fcof1o 7238 2fvidinvd 7241 pmtrfcnv 19204 qtophmeo 23119 fsovcnvd 42190 |
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