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Theorem f1cocnv1 6623
Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cocnv1 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))

Proof of Theorem f1cocnv1
StepHypRef Expression
1 f1f1orn 6605 . 2 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
2 f1ococnv1 6622 . 2 (𝐹:𝐴1-1-onto→ran 𝐹 → (𝐹𝐹) = ( I ↾ 𝐴))
31, 2syl 17 1 (𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538   I cid 5427  ccnv 5522  ran crn 5524  cres 5525  ccom 5527  1-1wf1 6325  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:  f1eqcocnv  7039  f1eqcocnvOLD  7040  domss2  8664  diophrw  39697
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