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Theorem f1ococnv2 6875
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 6855 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
2 fococnv2 6874 . 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 1536   I cid 5581  ccnv 5687  cres 5690  ccom 5692  ontowfo 6560  1-1-ontowf1o 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569
This theorem is referenced by:  f1ococnv1  6877  f1ocnvfv2  7296  mapen  9179  hashfacen  14489  setcinv  18143  catcisolem  18163  symginv  19434  f1omvdco2  19480  gsumval3  19939  gsumzf1o  19944  rngcinv  20653  ringcinv  20687  psrass1lem  21969  evl1var  22355  pf1ind  22374  fcobij  32739  symgfcoeu  33084  cycpmconjvlem  33143  cycpmconjs  33158  cyc3conja  33159  erdsze2lem2  35188  ltrncoidN  40110  cdlemg46  40717  cdlemk45  40929  cdlemk55a  40941  tendocnv  41003  eldioph2  42749  rngcinvALTV  48119  ringcinvALTV  48153
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