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

Theorem imacnvcnv 6225
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 6223 . . 3 (𝐴𝐵) = (𝐴𝐵)
21rneqi 5947 . 2 ran (𝐴𝐵) = ran (𝐴𝐵)
3 df-ima 5697 . 2 (𝐴𝐵) = ran (𝐴𝐵)
4 df-ima 5697 . 2 (𝐴𝐵) = ran (𝐴𝐵)
52, 3, 43eqtr4i 2774 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  ccnv 5683  ran crn 5685  cres 5686  cima 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697
This theorem is referenced by:  curry1  8130  curry2  8133  fnwelem  8157  fpwwe2lem5  10676  fpwwe2lem8  10679  eqglact  19198  hmeoima  23774  hmeocld  23776  hmeocls  23777  hmeontr  23778  reghmph  23802  qtopf1  23825  tgpconncompeqg  24121  imasf1obl  24502  mbfimaopnlem  25691  hmeoclda  36335
  Copyright terms: Public domain W3C validator