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Theorem imacnvcnv 5274
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 5272 . . 3  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
21rneqi 5036 . 2  |-  ran  ( `' `' A  |`  B )  =  ran  ( A  |`  B )
3 df-ima 4831 . 2  |-  ( `' `' A " B )  =  ran  ( `' `' A  |`  B )
4 df-ima 4831 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2417 1  |-  ( `' `' A " B )  =  ( A " B )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   `'ccnv 4817   ran crn 4819    |` cres 4820   "cima 4821
This theorem is referenced by:  curry1  6377  curry2  6380  fnwelem  6397  mapfien  7586  fpwwe2lem6  8443  fpwwe2lem9  8446  eqglact  14918  hmeoima  17718  hmeocld  17720  hmeocls  17721  hmeontr  17722  reghmph  17746  qtopf1  17769  tgpconcompeqg  18062  imasf1obl  18408  mbfimaopnlem  19414  hmeoclda  26027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-xp 4824  df-rel 4825  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831
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