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Theorem f1cocnv1 6079
 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 6061 . 2 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
2 f1ococnv1 6078 . 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 1475   I cid 4948  ◡ccnv 5037  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  –1-1→wf1 5801  –1-1-onto→wf1o 5803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811 This theorem is referenced by:  f1eqcocnv  6456  domss2  8004  diophrw  36340
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