Theorem imaeq1d 5105

Description: Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Hypothesis

Ref	Expression
imaeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
imaeq1d	|- ( ph -> ( A " C ) = ( B " C ) )

Proof of Theorem imaeq1d

Step	Hyp	Ref	Expression
1		imaeq1d.1	. 2 |- ( ph -> A = B )
2		imaeq1 5101	. 2 |- ( A = B -> ( A " C ) = ( B " C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A " C ) = ( B " C ) )

Syntax hints:   -> wi 4   = wceq 1398   "cima 4757

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767

This theorem is referenced by:  imaeq12d  5107  nfimad  5115  f1imacnv  5636  foimacnv  5637  suppssof1  6293  ssenen  7118  1arith  13090  ballotfilemscr  13206  ballotfilemrv  13207  ballotfilemrinv0  13220  eqglact  13978  psrbag  14943  psr1clfi  14969  iscn  15188