Theorem reseq2d 5058

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 5053	. 2 |- ( A = B -> ( C |` A ) = ( C |` B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C |` A ) = ( C |` B ) )

Syntax hints:   -> wi 4   = wceq 1402   |` cres 4771

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-opab 4188  df-xp 4775  df-res 4781

This theorem is referenced by:  reseq12d  5059  resima2  5092  relresfld  5312  f1orescnv  5650  funcocnv2  5659  fococnv2  5660  fnressn  5892  oprssov  6221  dftpos2  6522  fnsnsplitdc  6768  dif1en  7173  sbthlemi4  7267  fseq1p1m1  10479  resunimafz0  11252  setsvala  13361  gzsumsplit0  14125  metreslem  15404  xmspropd  15501  mspropd  15502  egrsubgr  16418  eupthvdres  16630  eupth2lem3fi  16631  eupth2fi  16634  bj-charfundcALT  16749