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Theorem f1cocnv1 6874
Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cocnv1 (𝐹:𝐴–1-1→𝐡 β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))

Proof of Theorem f1cocnv1
StepHypRef Expression
1 f1f1orn 6855 . 2 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
2 f1ococnv1 6873 . 2 (𝐹:𝐴–1-1-ontoβ†’ran 𝐹 β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))
31, 2syl 17 1 (𝐹:𝐴–1-1→𝐡 β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   I cid 5579  β—‘ccnv 5681  ran crn 5683   β†Ύ cres 5684   ∘ ccom 5686  β€“1-1β†’wf1 6550  β€“1-1-ontoβ†’wf1o 6552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560
This theorem is referenced by:  f1eqcocnv  7316  f1eqcocnvOLD  7317  domss2  9167  diophrw  42210
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