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Theorem imacnvcnv 5153
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 5151 . . 3  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
21rneqi 4921 . 2  |-  ran  ( `' `' A  |`  B )  =  ran  ( A  |`  B )
3 df-ima 4718 . 2  |-  ( `' `' A " B )  =  ran  ( `' `' A  |`  B )
4 df-ima 4718 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2326 1  |-  ( `' `' A " B )  =  ( A " B )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   `'ccnv 4704   ran crn 4706    |` cres 4707   "cima 4708
This theorem is referenced by:  curry1  6226  curry2  6229  fnwelem  6246  mapfien  7415  fpwwe2lem6  8273  fpwwe2lem9  8276  eqglact  14684  hmeoima  17472  hmeocld  17474  hmeocls  17475  hmeontr  17476  reghmph  17500  qtopf1  17523  tgpconcompeqg  17810  imasf1obl  18050  mbfimaopnlem  19026  f2imacnv  25155  domrancur1b  25303  hmeoclda  26354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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