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

Theorem funcocnv2 6074
 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 5806 . . 3 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
21simprbi 479 . 2 (Fun 𝐹 → (𝐹𝐹) ⊆ I )
3 iss 5367 . . 3 ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ dom (𝐹𝐹)))
4 dfdm4 5238 . . . . . . 7 dom 𝐹 = ran 𝐹
5 dmcoeq 5309 . . . . . . 7 (dom 𝐹 = ran 𝐹 → dom (𝐹𝐹) = dom 𝐹)
64, 5ax-mp 5 . . . . . 6 dom (𝐹𝐹) = dom 𝐹
7 df-rn 5049 . . . . . 6 ran 𝐹 = dom 𝐹
86, 7eqtr4i 2635 . . . . 5 dom (𝐹𝐹) = ran 𝐹
98reseq2i 5314 . . . 4 ( I ↾ dom (𝐹𝐹)) = ( I ↾ ran 𝐹)
109eqeq2i 2622 . . 3 ((𝐹𝐹) = ( I ↾ dom (𝐹𝐹)) ↔ (𝐹𝐹) = ( I ↾ ran 𝐹))
113, 10bitri 263 . 2 ((𝐹𝐹) ⊆ I ↔ (𝐹𝐹) = ( I ↾ ran 𝐹))
122, 11sylib 207 1 (Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ⊆ wss 3540   I cid 4948  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  Rel wrel 5043  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-fun 5806 This theorem is referenced by:  fococnv2  6075  f1cocnv2  6077  funcoeqres  6080  fcoinver  28798  cocnv  32690  frege131d  37075
 Copyright terms: Public domain W3C validator