Theorem imaeq2d 5022

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 5018	. 2 |- ( A = B -> ( C " A ) = ( C " B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1373   "cima 4678

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688

This theorem is referenced by:  imaeq12d  5023  nfimad  5031  elimasng  5050  ressn  5223  foima  5503  f1imacnv  5539  fvco2  5648  fsn2  5754  resfunexg  5805  funfvima3  5818  funiunfvdm  5832  isoselem  5889  fnexALT  6196  eceq1  6655  uniqs2  6682  ecinxp  6697  mapsn  6777  en2  6912  phplem4  6952  phplem4dom  6959  phplem4on  6964  sbthlem2  7060  isbth  7069  resunimafz0  10976  ennnfonelemg  12774  ennnfonelemhf1o  12784  ennnfonelemex  12785  ennnfonelemrn  12790  cnntr  14697  cnptopresti  14710  cnptoprest  14711