Theorem reseq2d 5065

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 5060	. 2 |- ( A = B -> ( C |` A ) = ( C |` B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C |` A ) = ( C |` B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   |` cres 4778

This proof depends on 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 proof 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 4194  df-xp 4782  df-res 4788

This theorem is used by:  reseq12d  5066  resima2  5099  relresfld  5320  f1orescnv  5658  funcocnv2  5667  fococnv2  5668  fnressn  5904  oprssov  6234  dftpos2  6536  fnsnsplitdc  6782  dif1en  7187  sbthlemi4  7282  fseq1p1m1  10518  resunimafz0  11296  setsvala  13441  gzsumsplit0  14238  gsumresfi  14263  gsum2dfilem2  14272  metreslem  15606  xmspropd  15703  mspropd  15704  egrsubgr  16709  eupthvdres  16921  eupth2lem3fi  16922  eupth2fi  16925  bj-charfundcALT  17040