Theorem imaeq2d 5023

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 5019	. 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 4679

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 4046  df-opab 4107  df-xp 4682  df-cnv 4684  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689

This theorem is referenced by:  imaeq12d  5024  nfimad  5032  elimasng  5051  ressn  5224  foima  5505  f1imacnv  5541  fvco2  5650  fsn2  5756  resfunexg  5807  funfvima3  5820  funiunfvdm  5834  isoselem  5891  fnexALT  6198  eceq1  6657  uniqs2  6684  ecinxp  6699  mapsn  6779  en2  6914  phplem4  6954  phplem4dom  6961  phplem4on  6966  sbthlem2  7062  isbth  7071  resunimafz0  10978  ennnfonelemg  12807  ennnfonelemhf1o  12817  ennnfonelemex  12818  ennnfonelemrn  12823  cnntr  14730  cnptopresti  14743  cnptoprest  14744