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

Theorem imacnvcnv 6205
Description: The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
Assertion
Ref Expression
imacnvcnv (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem imacnvcnv
StepHypRef Expression
1 rescnvcnv 6203 . . 3 (𝐴𝐵) = (𝐴𝐵)
21rneqi 5936 . 2 ran (𝐴𝐵) = ran (𝐴𝐵)
3 df-ima 5689 . 2 (𝐴𝐵) = ran (𝐴𝐵)
4 df-ima 5689 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2770 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  ccnv 5675  ran crn 5677  cres 5678  cima 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  curry1  8089  curry2  8092  fnwelem  8116  fpwwe2lem5  10629  fpwwe2lem8  10632  eqglact  19058  hmeoima  23268  hmeocld  23270  hmeocls  23271  hmeontr  23272  reghmph  23296  qtopf1  23319  tgpconncompeqg  23615  imasf1obl  23996  mbfimaopnlem  25171  hmeoclda  35213
  Copyright terms: Public domain W3C validator