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Theorem funcocnv2 5440
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 5173 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
21simprbi 273 . 2 (Fun 𝐹 → (𝐹𝐹) ⊆ I )
3 iss 4913 . . 3 ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ dom (𝐹𝐹)))
4 dfdm4 4779 . . . . . . . 8 dom 𝐹 = ran 𝐹
5 dmcoeq 4859 . . . . . . . 8 (dom 𝐹 = ran 𝐹 → dom (𝐹𝐹) = dom 𝐹)
64, 5ax-mp 5 . . . . . . 7 dom (𝐹𝐹) = dom 𝐹
7 df-rn 4598 . . . . . . 7 ran 𝐹 = dom 𝐹
86, 7eqtr4i 2181 . . . . . 6 dom (𝐹𝐹) = ran 𝐹
98a1i 9 . . . . 5 (Fun 𝐹 → dom (𝐹𝐹) = ran 𝐹)
109reseq2d 4867 . . . 4 (Fun 𝐹 → ( I ↾ dom (𝐹𝐹)) = ( I ↾ ran 𝐹))
1110eqeq2d 2169 . . 3 (Fun 𝐹 → ((𝐹𝐹) = ( I ↾ dom (𝐹𝐹)) ↔ (𝐹𝐹) = ( I ↾ ran 𝐹)))
123, 11syl5bb 191 . 2 (Fun 𝐹 → ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ ran 𝐹)))
132, 12mpbid 146 1 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wss 3102   I cid 4249  ccnv 4586  dom cdm 4587  ran crn 4588  cres 4589  ccom 4591  Rel wrel 4592  Fun wfun 5165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-fun 5173
This theorem is referenced by:  fococnv2  5441  f1cocnv2  5443  funcoeqres  5446
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