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Theorem imaeq12d 5144
Description: Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
Hypotheses
Ref Expression
imaeq1d.1  |-  ( ph  ->  A  =  B )
imaeq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
imaeq12d  |-  ( ph  ->  ( A " C
)  =  ( B
" D ) )

Proof of Theorem imaeq12d
StepHypRef Expression
1 imaeq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21imaeq1d 5142 . 2  |-  ( ph  ->  ( A " C
)  =  ( B
" C ) )
3 imaeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43imaeq2d 5143 . 2  |-  ( ph  ->  ( B " C
)  =  ( B
" D ) )
52, 4eqtrd 2419 1  |-  ( ph  ->  ( A " C
)  =  ( B
" D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   "cima 4821
This theorem is referenced by:  csbima12g  5153  vdwpc  13275  dmdprd  15486  isunit  15689  qtopval  17648  limciun  19648  ig1pval  19962  ispth  21422  qqhval  24157  orvcval  24494  ballotlemrval  24554  ballotlemrinv0  24569  ballotlemrinv  24570  itg2addnclem2  25958  islmodfg  26836
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
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-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  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-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831
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