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Theorem fococnv2 6704
Description: The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
Assertion
Ref Expression
fococnv2 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))

Proof of Theorem fococnv2
StepHypRef Expression
1 fofun 6652 . . 3 (𝐹:𝐴onto𝐵 → Fun 𝐹)
2 funcocnv2 6703 . . 3 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
31, 2syl 17 . 2 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))
4 forn 6654 . . 3 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
54reseq2d 5865 . 2 (𝐹:𝐴onto𝐵 → ( I ↾ ran 𝐹) = ( I ↾ 𝐵))
63, 5eqtrd 2778 1 (𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543   I cid 5468  ccnv 5564  ran crn 5566  cres 5567  ccom 5569  Fun wfun 6391  ontowfo 6395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-br 5068  df-opab 5130  df-id 5469  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-fun 6399  df-fn 6400  df-f 6401  df-fo 6403
This theorem is referenced by:  f1ococnv2  6705  foeqcnvco  7128
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