Theorem imaeq2d 5041

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 5037	. 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 4696

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by:  imaeq12d  5042  nfimad  5050  elimasng  5069  ressn  5242  foima  5525  f1imacnv  5561  fvco2  5671  fsn2  5777  resfunexg  5828  funfvima3  5841  funiunfvdm  5855  isoselem  5912  fnexALT  6219  eceq1  6678  uniqs2  6705  ecinxp  6720  mapsn  6800  en2  6936  phplem4  6977  phplem4dom  6984  phplem4on  6990  sbthlem2  7086  isbth  7095  resunimafz0  11013  ennnfonelemg  12889  ennnfonelemhf1o  12899  ennnfonelemex  12900  ennnfonelemrn  12905  cnntr  14812  cnptopresti  14825  cnptoprest  14826