Theorem imaeq2d 5006

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 5002	. 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 4663

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673

This theorem is referenced by:  imaeq12d  5007  nfimad  5015  elimasng  5034  ressn  5207  foima  5482  f1imacnv  5518  fvco2  5627  fsn2  5733  resfunexg  5780  funfvima3  5793  funiunfvdm  5807  isoselem  5864  fnexALT  6165  eceq1  6624  uniqs2  6651  ecinxp  6666  mapsn  6746  phplem4  6913  phplem4dom  6920  phplem4on  6925  sbthlem2  7019  isbth  7028  resunimafz0  10905  ennnfonelemg  12563  ennnfonelemhf1o  12573  ennnfonelemex  12574  ennnfonelemrn  12579  cnntr  14404  cnptopresti  14417  cnptoprest  14418