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Theorem funcocnv2 5358
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 5093 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
21simprbi 271 . 2 (Fun 𝐹 → (𝐹𝐹) ⊆ I )
3 iss 4833 . . 3 ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ dom (𝐹𝐹)))
4 dfdm4 4699 . . . . . . . 8 dom 𝐹 = ran 𝐹
5 dmcoeq 4779 . . . . . . . 8 (dom 𝐹 = ran 𝐹 → dom (𝐹𝐹) = dom 𝐹)
64, 5ax-mp 5 . . . . . . 7 dom (𝐹𝐹) = dom 𝐹
7 df-rn 4518 . . . . . . 7 ran 𝐹 = dom 𝐹
86, 7eqtr4i 2139 . . . . . 6 dom (𝐹𝐹) = ran 𝐹
98a1i 9 . . . . 5 (Fun 𝐹 → dom (𝐹𝐹) = ran 𝐹)
109reseq2d 4787 . . . 4 (Fun 𝐹 → ( I ↾ dom (𝐹𝐹)) = ( I ↾ ran 𝐹))
1110eqeq2d 2127 . . 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 1314  wss 3039   I cid 4178  ccnv 4506  dom cdm 4507  ran crn 4508  cres 4509  ccom 4511  Rel wrel 4512  Fun wfun 5085
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-fun 5093
This theorem is referenced by:  fococnv2  5359  f1cocnv2  5361  funcoeqres  5364
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