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Theorem imaeq12d 4885
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 4883 . 2  |-  ( ph  ->  ( A " C
)  =  ( B
" C ) )
3 imaeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43imaeq2d 4884 . 2  |-  ( ph  ->  ( B " C
)  =  ( B
" D ) )
52, 4eqtrd 2172 1  |-  ( ph  ->  ( A " C
)  =  ( B
" D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   "cima 4545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3933  df-opab 3993  df-xp 4548  df-cnv 4550  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555
This theorem is referenced by:  csbima12g  4903  caseinl  6979  caseinr  6980
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