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Theorem imaeq12d 5013
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 5011 . 2  |-  ( ph  ->  ( A " C
)  =  ( B
" C ) )
3 imaeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43imaeq2d 5012 . 2  |-  ( ph  ->  ( B " C
)  =  ( B
" D ) )
52, 4eqtrd 2315 1  |-  ( ph  ->  ( A " C
)  =  ( B
" D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   "cima 4692
This theorem is referenced by:  csbima12g  5022  limciun  19244  ballotlemrval  23076  ballotlemrinv0  23091  ballotlemrinv  23092
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  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-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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