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Theorem funcocnv2 6859
Description: Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
funcocnv2 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))

Proof of Theorem funcocnv2
StepHypRef Expression
1 df-fun 6546 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
21simprbi 498 . 2 (Fun 𝐹 → (𝐹𝐹) ⊆ I )
3 iss 6036 . . 3 ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ dom (𝐹𝐹)))
4 dfdm4 5896 . . . . . . 7 dom 𝐹 = ran 𝐹
5 dmcoeq 5974 . . . . . . 7 (dom 𝐹 = ran 𝐹 → dom (𝐹𝐹) = dom 𝐹)
64, 5ax-mp 5 . . . . . 6 dom (𝐹𝐹) = dom 𝐹
7 df-rn 5688 . . . . . 6 ran 𝐹 = dom 𝐹
86, 7eqtr4i 2764 . . . . 5 dom (𝐹𝐹) = ran 𝐹
98reseq2i 5979 . . . 4 ( I ↾ dom (𝐹𝐹)) = ( I ↾ ran 𝐹)
109eqeq2i 2746 . . 3 ((𝐹𝐹) = ( I ↾ dom (𝐹𝐹)) ↔ (𝐹𝐹) = ( I ↾ ran 𝐹))
113, 10bitri 275 . 2 ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ ran 𝐹))
122, 11sylib 217 1 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wss 3949   I cid 5574  ccnv 5676  dom cdm 5677  ran crn 5678  cres 5679  ccom 5681  Rel wrel 5682  Fun wfun 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-fun 6546
This theorem is referenced by:  fococnv2  6860  f1cocnv2  6862  funcoeqres  6865  fcoinver  31835  tocyc01  32277  cocnv  36593  frege131d  42515  isomushgr  46494
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