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Theorem imaeq12d 4971
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 4969 . 2  |-  ( ph  ->  ( A " C
)  =  ( B
" C ) )
3 imaeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43imaeq2d 4970 . 2  |-  ( ph  ->  ( B " C
)  =  ( B
" D ) )
52, 4eqtrd 2210 1  |-  ( ph  ->  ( A " C
)  =  ( B
" D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   "cima 4629
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639
This theorem is referenced by:  csbima12g  4989  caseinl  7089  caseinr  7090  isunitd  13273
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