MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcocnv2 Structured version   Visualization version   GIF version

Theorem funcocnv2 6859
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 6545 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
21simprbi 495 . 2 (Fun 𝐹 → (𝐹𝐹) ⊆ I )
3 iss 6034 . . 3 ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ dom (𝐹𝐹)))
4 dfdm4 5892 . . . . . . 7 dom 𝐹 = ran 𝐹
5 dmcoeq 5971 . . . . . . 7 (dom 𝐹 = ran 𝐹 → dom (𝐹𝐹) = dom 𝐹)
64, 5ax-mp 5 . . . . . 6 dom (𝐹𝐹) = dom 𝐹
7 df-rn 5683 . . . . . 6 ran 𝐹 = dom 𝐹
86, 7eqtr4i 2756 . . . . 5 dom (𝐹𝐹) = ran 𝐹
98reseq2i 5976 . . . 4 ( I ↾ dom (𝐹𝐹)) = ( I ↾ ran 𝐹)
109eqeq2i 2738 . . 3 ((𝐹𝐹) = ( I ↾ dom (𝐹𝐹)) ↔ (𝐹𝐹) = ( I ↾ ran 𝐹))
113, 10bitri 274 . 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 1533  wss 3939   I cid 5569  ccnv 5671  dom cdm 5672  ran crn 5673  cres 5674  ccom 5676  Rel wrel 5677  Fun wfun 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-fun 6545
This theorem is referenced by:  fococnv2  6860  f1cocnv2  6862  funcoeqres  6865  fcoinver  32439  tocyc01  32884  cocnv  37255  frege131d  43259  gricushgr  47295
  Copyright terms: Public domain W3C validator