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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 6646 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵) | |
2 | fococnv2 6664 | . 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 1543 I cid 5439 ◡ccnv 5535 ↾ cres 5538 ∘ ccom 5540 –onto→wfo 6356 –1-1-onto→wf1o 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 |
This theorem is referenced by: f1ococnv1 6667 f1ocnvfv2 7066 mapen 8788 hashfacen 13983 hashfacenOLD 13984 setcinv 17550 catcisolem 17570 symginv 18748 f1omvdco2 18794 gsumval3 19246 gsumzf1o 19251 psrass1lemOLD 20853 psrass1lem 20856 evl1var 21206 pf1ind 21225 fcobij 30731 symgfcoeu 31024 cycpmconjvlem 31081 cycpmconjs 31096 cyc3conja 31097 erdsze2lem2 32833 ltrncoidN 37828 cdlemg46 38435 cdlemk45 38647 cdlemk55a 38659 tendocnv 38721 eldioph2 40228 rngcinv 45155 rngcinvALTV 45167 ringcinv 45206 ringcinvALTV 45230 |
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