Theorem reseq1i 4921

Description: Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)

Hypothesis

Ref	Expression
reseqi.1	|- A = B

Assertion

Ref	Expression
reseq1i	|- ( A |` C ) = ( B |` C )

Proof of Theorem reseq1i

Step	Hyp	Ref	Expression
1		reseqi.1	. 2 |- A = B
2		reseq1 4919	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1364   |` cres 4646

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-res 4656

This theorem is referenced by:  reseq12i  4923  resindm  4967  resmpt  4973  resmpt3  4974  resmptf  4975  opabresid  4978  rescnvcnv  5109  coires1  5164  fcoi1  5415  fvsnun1  5733  fvsnun2  5734  resoprab  5991  resmpo  5993  ofmres  6160  f1stres  6183  f2ndres  6184  df1st2  6243  df2nd2  6244  dftpos2  6285  tfr2a  6345  freccllem  6426  frecfcllem  6428  frecsuclem  6430  djuf1olemr  7082  divfnzn  9650  cnmptid  14233  xmsxmet2  14415  msmet2  14416