Theorem imaeq12d 5022

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

Step	Hyp	Ref	Expression
1		imaeq1d.1	. . 3 |- ( ph -> A = B )
2	1	imaeq1d 5020	. 2 |- ( ph -> ( A " C ) = ( B " C ) )
3		imaeq12d.2	. . 3 |- ( ph -> C = D )
4	3	imaeq2d 5021	. 2 |- ( ph -> ( B " C ) = ( B " D ) )
5	2, 4	eqtrd 2237	1 |- ( ph -> ( A " C ) = ( B " D ) )

Syntax hints:   -> wi 4   = wceq 1372   "cima 4677

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-cnv 4682  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687

This theorem is referenced by:  csbima12g  5042  caseinl  7192  caseinr  7193  isunitd  13839