Theorem imaeq1d 4667

Description: Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Hypothesis

Ref	Expression
imaeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
imaeq1d	|- ( ph -> ( A " C ) = ( B " C ) )

Proof of Theorem imaeq1d

Step	Hyp	Ref	Expression
1		imaeq1d.1	. 2 |- ( ph -> A = B )
2		imaeq1 4663	. 2 |- ( A = B -> ( A " C ) = ( B " C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A " C ) = ( B " C ) )

Syntax hints:   -> wi 4   = wceq 1243   "cima 4348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358

This theorem is referenced by:  imaeq12d  4669  nfimad  4677  f1imacnv  5143  foimacnv  5144  suppssof1  5728