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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 6791 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵) | |
2 | fococnv2 6810 | . 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 1541 I cid 5530 ◡ccnv 5632 ↾ cres 5635 ∘ ccom 5637 –onto→wfo 6494 –1-1-onto→wf1o 6495 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
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-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 |
This theorem is referenced by: f1ococnv1 6813 f1ocnvfv2 7222 mapen 9084 hashfacen 14350 hashfacenOLD 14351 setcinv 17975 catcisolem 17995 symginv 19182 f1omvdco2 19228 gsumval3 19682 gsumzf1o 19687 psrass1lemOLD 21340 psrass1lem 21343 evl1var 21700 pf1ind 21719 fcobij 31583 symgfcoeu 31877 cycpmconjvlem 31934 cycpmconjs 31949 cyc3conja 31950 erdsze2lem2 33738 ltrncoidN 38581 cdlemg46 39188 cdlemk45 39400 cdlemk55a 39412 tendocnv 39474 eldioph2 41062 rngcinv 46250 rngcinvALTV 46262 ringcinv 46301 ringcinvALTV 46325 |
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