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Theorem f1ococnv2 6120
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.)
Assertion
Ref Expression
f1ococnv2 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))

Proof of Theorem f1ococnv2
StepHypRef Expression
1 f1ofo 6101 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
2 fococnv2 6119 . 2 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
31, 2syl 17 1 (𝐹:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480   I cid 4984  ccnv 5073  cres 5076  ccom 5078  ontowfo 5845  1-1-ontowf1o 5846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854
This theorem is referenced by:  f1ococnv1  6122  f1ocnvfv2  6487  mapen  8068  hashfacen  13176  setcinv  16661  catcisolem  16677  symginv  17743  f1omvdco2  17789  gsumval3  18229  gsumzf1o  18234  psrass1lem  19296  evl1var  19619  pf1ind  19638  fcobij  29343  symgfcoeu  29630  erdsze2lem2  30894  ltrncoidN  34894  cdlemg46  35503  cdlemk45  35715  cdlemk55a  35727  tendocnv  35790  eldioph2  36805  rngcinv  41269  rngcinvALTV  41281  ringcinv  41320  ringcinvALTV  41344
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