Theorem imaeq1d 4697

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 4693	. 2 |- ( A = B -> ( A " C ) = ( B " C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A " C ) = ( B " C ) )

Syntax hints:   -> wi 4   = wceq 1285   "cima 4374

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384

This theorem is referenced by:  imaeq12d  4699  nfimad  4707  f1imacnv  5174  foimacnv  5175  suppssof1  5759