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Theorem f1cocnv2 5488
Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cocnv2 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))

Proof of Theorem f1cocnv2
StepHypRef Expression
1 f1fun 5423 . 2 (𝐹:𝐴1-1𝐵 → Fun 𝐹)
2 funcocnv2 5485 . 2 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
31, 2syl 14 1 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353   I cid 4287  ccnv 4624  ran crn 4626  cres 4627  ccom 4629  Fun wfun 5209  1-1wf1 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220
This theorem is referenced by: (None)
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