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Theorem imacnvcnv 5327
Description: The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
Assertion
Ref Expression
imacnvcnv  |-  ( `' `' A " B )  =  ( A " B )

Proof of Theorem imacnvcnv
StepHypRef Expression
1 rescnvcnv 5325 . . 3  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
21rneqi 5089 . 2  |-  ran  ( `' `' A  |`  B )  =  ran  ( A  |`  B )
3 df-ima 4884 . 2  |-  ( `' `' A " B )  =  ran  ( `' `' A  |`  B )
4 df-ima 4884 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2466 1  |-  ( `' `' A " B )  =  ( A " B )
Colors of variables: wff set class
Syntax hints:    = wceq 1652   `'ccnv 4870   ran crn 4872    |` cres 4873   "cima 4874
This theorem is referenced by:  curry1  6431  curry2  6434  fnwelem  6454  mapfien  7646  fpwwe2lem6  8503  fpwwe2lem9  8506  eqglact  14984  hmeoima  17790  hmeocld  17792  hmeocls  17793  hmeontr  17794  reghmph  17818  qtopf1  17841  tgpconcompeqg  18134  imasf1obl  18511  mbfimaopnlem  19540  hmeoclda  26328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-xp 4877  df-rel 4878  df-cnv 4879  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884
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