Theorem imaeq12d 4818

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 4816	. 2 |- ( ph -> ( A " C ) = ( B " C ) )
3		imaeq12d.2	. . 3 |- ( ph -> C = D )
4	3	imaeq2d 4817	. 2 |- ( ph -> ( B " C ) = ( B " D ) )
5	2, 4	eqtrd 2132	1 |- ( ph -> ( A " C ) = ( B " D ) )

Syntax hints:   -> wi 4   = wceq 1299   "cima 4480

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490

This theorem is referenced by:  csbima12g  4836  caseinl  6891  caseinr  6892