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Theorem funcocnv2 5178
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 4931 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
21simprbi 264 . 2 (Fun 𝐹 → (𝐹𝐹) ⊆ I )
3 iss 4681 . . 3 ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ dom (𝐹𝐹)))
4 dfdm4 4554 . . . . . . . 8 dom 𝐹 = ran 𝐹
5 dmcoeq 4631 . . . . . . . 8 (dom 𝐹 = ran 𝐹 → dom (𝐹𝐹) = dom 𝐹)
64, 5ax-mp 7 . . . . . . 7 dom (𝐹𝐹) = dom 𝐹
7 df-rn 4383 . . . . . . 7 ran 𝐹 = dom 𝐹
86, 7eqtr4i 2079 . . . . . 6 dom (𝐹𝐹) = ran 𝐹
98a1i 9 . . . . 5 (Fun 𝐹 → dom (𝐹𝐹) = ran 𝐹)
109reseq2d 4639 . . . 4 (Fun 𝐹 → ( I ↾ dom (𝐹𝐹)) = ( I ↾ ran 𝐹))
1110eqeq2d 2067 . . 3 (Fun 𝐹 → ((𝐹𝐹) = ( I ↾ dom (𝐹𝐹)) ↔ (𝐹𝐹) = ( I ↾ ran 𝐹)))
123, 11syl5bb 185 . 2 (Fun 𝐹 → ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ ran 𝐹)))
132, 12mpbid 139 1 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wss 2944   I cid 4052  ccnv 4371  dom cdm 4372  ran crn 4373  cres 4374  ccom 4376  Rel wrel 4377  Fun wfun 4923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-fun 4931
This theorem is referenced by:  fococnv2  5179  f1cocnv2  5181  funcoeqres  5184
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