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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 1540   I cid 5577  ccnv 5684  cres 5687  ccom 5689  ontowfo 6559  1-1-ontowf1o 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568
This theorem is referenced by:  f1ococnv1  6877  f1ocnvfv2  7297  mapen  9181  hashfacen  14493  setcinv  18135  catcisolem  18155  symginv  19420  f1omvdco2  19466  gsumval3  19925  gsumzf1o  19930  rngcinv  20637  ringcinv  20671  psrass1lem  21952  evl1var  22340  pf1ind  22359  fcobij  32733  symgfcoeu  33102  cycpmconjvlem  33161  cycpmconjs  33176  cyc3conja  33177  erdsze2lem2  35209  ltrncoidN  40130  cdlemg46  40737  cdlemk45  40949  cdlemk55a  40961  tendocnv  41023  eldioph2  42773  rngcinvALTV  48192  ringcinvALTV  48226
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