Theorem imaeq2d 4837

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 4833	. 2 |- ( A = B -> ( C " A ) = ( C " B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1312   "cima 4500

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 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-xp 4503  df-cnv 4505  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510

This theorem is referenced by:  imaeq12d  4838  nfimad  4846  elimasng  4863  ressn  5035  foima  5306  f1imacnv  5338  fvco2  5442  fsn2  5546  resfunexg  5593  funfvima3  5603  funiunfvdm  5616  isoselem  5673  fnexALT  5962  eceq1  6415  uniqs2  6440  ecinxp  6455  mapsn  6535  phplem4  6699  phplem4dom  6706  phplem4on  6711  sbthlem2  6795  isbth  6804  resunimafz0  10460  ennnfonelemg  11754  ennnfonelemhf1o  11764  ennnfonelemex  11765  ennnfonelemrn  11770  cnntr  12229  cnptopresti  12242  cnptoprest  12243