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Mirrors > Home > MPE Home > Th. List > f1ococnv2 | Structured version Visualization version GIF version |
Description: The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.) |
Ref | Expression |
---|---|
f1ococnv2 | ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ofo 6841 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵) | |
2 | fococnv2 6860 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 I cid 5574 ◡ccnv 5676 ↾ cres 5679 ∘ ccom 5681 –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-clab 2711 df-cleq 2725 df-clel 2811 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-br 5150 df-opab 5212 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-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 |
This theorem is referenced by: f1ococnv1 6863 f1ocnvfv2 7275 mapen 9141 hashfacen 14413 hashfacenOLD 14414 setcinv 18040 catcisolem 18060 symginv 19270 f1omvdco2 19316 gsumval3 19775 gsumzf1o 19780 psrass1lemOLD 21493 psrass1lem 21496 evl1var 21855 pf1ind 21874 fcobij 31947 symgfcoeu 32243 cycpmconjvlem 32300 cycpmconjs 32315 cyc3conja 32316 erdsze2lem2 34195 ltrncoidN 38999 cdlemg46 39606 cdlemk45 39818 cdlemk55a 39830 tendocnv 39892 eldioph2 41500 rngcinv 46879 rngcinvALTV 46891 ringcinv 46930 ringcinvALTV 46954 |
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