Theorem reseq2d 5013

Description: Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

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

Assertion

Ref	Expression
reseq2d	|- ( ph -> ( C |` A ) = ( C |` B ) )

Proof of Theorem reseq2d

Step	Hyp	Ref	Expression
1		reseqd.1	. 2 |- ( ph -> A = B )
2		reseq2 5008	. 2 |- ( A = B -> ( C |` A ) = ( C |` B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1397   |` cres 4727

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-opab 4151  df-xp 4731  df-res 4737

This theorem is referenced by:  reseq12d  5014  resima2  5047  relresfld  5266  f1orescnv  5599  funcocnv2  5608  fococnv2  5609  fnressn  5839  oprssov  6163  dftpos2  6426  fnsnsplitdc  6672  dif1en  7067  sbthlemi4  7158  fseq1p1m1  10328  resunimafz0  11094  setsvala  13112  metreslem  15103  xmspropd  15200  mspropd  15201  egrsubgr  16113  bj-charfundcALT  16404