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Theorem imacnvcnv 5137
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 5135 . . 3  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
21rneqi 4905 . 2  |-  ran  ( `' `' A  |`  B )  =  ran  ( A  |`  B )
3 df-ima 4702 . 2  |-  ( `' `' A " B )  =  ran  ( `' `' A  |`  B )
4 df-ima 4702 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2313 1  |-  ( `' `' A " B )  =  ( A " B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692
This theorem is referenced by:  curry1  6210  curry2  6213  fnwelem  6230  mapfien  7399  fpwwe2lem6  8257  fpwwe2lem9  8260  eqglact  14668  hmeoima  17456  hmeocld  17458  hmeocls  17459  hmeontr  17460  reghmph  17484  qtopf1  17507  tgpconcompeqg  17794  imasf1obl  18034  mbfimaopnlem  19010  f2imacnv  25052  domrancur1b  25200  hmeoclda  26251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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