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Theorem f1ococnv2 6860
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 6840 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
2 fococnv2 6859 . 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 1541   I cid 5573  ccnv 5675  cres 5678  ccom 5680  ontowfo 6541  1-1-ontowf1o 6542
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550
This theorem is referenced by:  f1ococnv1  6862  f1ocnvfv2  7277  mapen  9143  hashfacen  14415  hashfacenOLD  14416  setcinv  18042  catcisolem  18062  symginv  19272  f1omvdco2  19318  gsumval3  19777  gsumzf1o  19782  psrass1lemOLD  21499  psrass1lem  21502  evl1var  21862  pf1ind  21881  fcobij  31985  symgfcoeu  32284  cycpmconjvlem  32341  cycpmconjs  32356  cyc3conja  32357  erdsze2lem2  34264  ltrncoidN  39085  cdlemg46  39692  cdlemk45  39904  cdlemk55a  39916  tendocnv  39978  eldioph2  41582  rngcinv  46958  rngcinvALTV  46970  ringcinv  47009  ringcinvALTV  47033
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