Theorem imaeq2d 5009

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 5005	. 2 |- ( A = B -> ( C " A ) = ( C " B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1364   "cima 4666

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676

This theorem is referenced by:  imaeq12d  5010  nfimad  5018  elimasng  5037  ressn  5210  foima  5485  f1imacnv  5521  fvco2  5630  fsn2  5736  resfunexg  5783  funfvima3  5796  funiunfvdm  5810  isoselem  5867  fnexALT  6168  eceq1  6627  uniqs2  6654  ecinxp  6669  mapsn  6749  phplem4  6916  phplem4dom  6923  phplem4on  6928  sbthlem2  7024  isbth  7033  resunimafz0  10923  ennnfonelemg  12620  ennnfonelemhf1o  12630  ennnfonelemex  12631  ennnfonelemrn  12636  cnntr  14461  cnptopresti  14474  cnptoprest  14475