Theorem imaeq2d 5068

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

Hypothesis

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

Assertion

Ref	Expression
imaeq2d	|- ( ph -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2d

Step	Hyp	Ref	Expression
1		imaeq1d.1	. 2 |- ( ph -> A = B )
2		imaeq2 5064	. 2 |- ( A = B -> ( C " A ) = ( C " B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1395   "cima 4722

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by:  imaeq12d  5069  nfimad  5077  elimasng  5096  ressn  5269  foima  5553  f1imacnv  5589  fvco2  5703  fsn2  5809  resfunexg  5860  funfvima3  5873  funiunfvdm  5887  isoselem  5944  fnexALT  6256  eceq1  6715  uniqs2  6742  ecinxp  6757  mapsn  6837  en2  6973  phplem4  7016  phplem4dom  7023  phplem4on  7029  sbthlem2  7125  isbth  7134  resunimafz0  11053  ennnfonelemg  12974  ennnfonelemhf1o  12984  ennnfonelemex  12985  ennnfonelemrn  12990  cnntr  14899  cnptopresti  14912  cnptoprest  14913