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Theorem funcocnv2 6626
 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 6337 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
21simprbi 500 . 2 (Fun 𝐹 → (𝐹𝐹) ⊆ I )
3 iss 5875 . . 3 ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ dom (𝐹𝐹)))
4 dfdm4 5735 . . . . . . 7 dom 𝐹 = ran 𝐹
5 dmcoeq 5815 . . . . . . 7 (dom 𝐹 = ran 𝐹 → dom (𝐹𝐹) = dom 𝐹)
64, 5ax-mp 5 . . . . . 6 dom (𝐹𝐹) = dom 𝐹
7 df-rn 5535 . . . . . 6 ran 𝐹 = dom 𝐹
86, 7eqtr4i 2784 . . . . 5 dom (𝐹𝐹) = ran 𝐹
98reseq2i 5820 . . . 4 ( I ↾ dom (𝐹𝐹)) = ( I ↾ ran 𝐹)
109eqeq2i 2771 . . 3 ((𝐹𝐹) = ( I ↾ dom (𝐹𝐹)) ↔ (𝐹𝐹) = ( I ↾ ran 𝐹))
113, 10bitri 278 . 2 ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ ran 𝐹))
122, 11sylib 221 1 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ⊆ wss 3858   I cid 5429  ◡ccnv 5523  dom cdm 5524  ran crn 5525   ↾ cres 5526   ∘ ccom 5528  Rel wrel 5529  Fun wfun 6329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-fun 6337 This theorem is referenced by:  fococnv2  6627  f1cocnv2  6629  funcoeqres  6632  fcoinver  30468  tocyc01  30911  cocnv  35465  frege131d  40860  isomushgr  44733
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