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Theorem funcocnv2 6814
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 6503 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
21simprbi 498 . 2 (Fun 𝐹 → (𝐹𝐹) ⊆ I )
3 iss 5994 . . 3 ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ dom (𝐹𝐹)))
4 dfdm4 5856 . . . . . . 7 dom 𝐹 = ran 𝐹
5 dmcoeq 5934 . . . . . . 7 (dom 𝐹 = ran 𝐹 → dom (𝐹𝐹) = dom 𝐹)
64, 5ax-mp 5 . . . . . 6 dom (𝐹𝐹) = dom 𝐹
7 df-rn 5649 . . . . . 6 ran 𝐹 = dom 𝐹
86, 7eqtr4i 2768 . . . . 5 dom (𝐹𝐹) = ran 𝐹
98reseq2i 5939 . . . 4 ( I ↾ dom (𝐹𝐹)) = ( I ↾ ran 𝐹)
109eqeq2i 2750 . . 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 3915   I cid 5535  ccnv 5637  dom cdm 5638  ran crn 5639  cres 5640  ccom 5642  Rel wrel 5643  Fun wfun 6495
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-fun 6503
This theorem is referenced by:  fococnv2  6815  f1cocnv2  6817  funcoeqres  6820  fcoinver  31567  tocyc01  32009  cocnv  36213  frege131d  42110  isomushgr  46092
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