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Theorem imacnvcnv 5107
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 5105 . . 3  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
21rneqi 4869 . 2  |-  ran  ( `' `' A  |`  B )  =  ran  ( A  |`  B )
3 df-ima 4653 . 2  |-  ( `' `' A " B )  =  ran  ( `' `' A  |`  B )
4 df-ima 4653 . 2  |-  ( A
" B )  =  ran  ( A  |`  B )
52, 3, 43eqtr4i 2219 1  |-  ( `' `' A " B )  =  ( A " B )
Colors of variables: wff set class
Syntax hints:    = wceq 1363   `'ccnv 4639   ran crn 4641    |` cres 4642   "cima 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018  df-opab 4079  df-xp 4646  df-rel 4647  df-cnv 4648  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653
This theorem is referenced by:  casedm  7102  caseinl  7107  caseinr  7108  djudm  7121  eqglact  13129  hmeoima  14193  hmeocld  14195  hmeontr  14196
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