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Theorem f1ococnv2 6811
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 6791 . 2 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
2 fococnv2 6810 . 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 5530  ccnv 5632  cres 5635  ccom 5637  ontowfo 6494  1-1-ontowf1o 6495
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
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 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503
This theorem is referenced by:  f1ococnv1  6813  f1ocnvfv2  7222  mapen  9084  hashfacen  14350  hashfacenOLD  14351  setcinv  17975  catcisolem  17995  symginv  19182  f1omvdco2  19228  gsumval3  19682  gsumzf1o  19687  psrass1lemOLD  21340  psrass1lem  21343  evl1var  21700  pf1ind  21719  fcobij  31583  symgfcoeu  31877  cycpmconjvlem  31934  cycpmconjs  31949  cyc3conja  31950  erdsze2lem2  33738  ltrncoidN  38581  cdlemg46  39188  cdlemk45  39400  cdlemk55a  39412  tendocnv  39474  eldioph2  41062  rngcinv  46250  rngcinvALTV  46262  ringcinv  46301  ringcinvALTV  46325
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