Theorem imaeq12d 4972

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 4970	. 2 |- ( ph -> ( A " C ) = ( B " C ) )
3		imaeq12d.2	. . 3 |- ( ph -> C = D )
4	3	imaeq2d 4971	. 2 |- ( ph -> ( B " C ) = ( B " D ) )
5	2, 4	eqtrd 2210	1 |- ( ph -> ( A " C ) = ( B " D ) )

Syntax hints:   -> wi 4   = wceq 1353   "cima 4630

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 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-cnv 4635  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640

This theorem is referenced by:  csbima12g  4990  caseinl  7090  caseinr  7091  isunitd  13275