Theorem imaeq12d 4877

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 4875	. 2 |- ( ph -> ( A " C ) = ( B " C ) )
3		imaeq12d.2	. . 3 |- ( ph -> C = D )
4	3	imaeq2d 4876	. 2 |- ( ph -> ( B " C ) = ( B " D ) )
5	2, 4	eqtrd 2170	1 |- ( ph -> ( A " C ) = ( B " D ) )

Syntax hints:   -> wi 4   = wceq 1331   "cima 4537

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 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547

This theorem is referenced by:  csbima12g  4895  caseinl  6969  caseinr  6970