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Theorem f1ococnv2 6620
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 6601 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
2 fococnv2 6619 . 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 1538   I cid 5427  ccnv 5522  cres 5525  ccom 5527  ontowfo 6326  1-1-ontowf1o 6327
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335
This theorem is referenced by:  f1ococnv1  6622  f1ocnvfv2  7016  mapen  8669  hashfacen  13812  setcinv  17345  catcisolem  17361  symginv  18525  f1omvdco2  18571  gsumval3  19023  gsumzf1o  19028  psrass1lem  20618  evl1var  20963  pf1ind  20982  fcobij  30487  symgfcoeu  30779  cycpmconjvlem  30836  cycpmconjs  30851  cyc3conja  30852  erdsze2lem2  32559  ltrncoidN  37417  cdlemg46  38024  cdlemk45  38236  cdlemk55a  38248  tendocnv  38310  eldioph2  39690  rngcinv  44592  rngcinvALTV  44604  ringcinv  44643  ringcinvALTV  44667
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