Theorem reseq1i 4810

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 4808	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1331   |` cres 4536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-res 4546

This theorem is referenced by:  reseq12i  4812  resindm  4856  resmpt  4862  resmpt3  4863  resmptf  4864  opabresid  4867  rescnvcnv  4996  coires1  5051  fcoi1  5298  fvsnun1  5610  fvsnun2  5611  resoprab  5860  resmpo  5862  ofmres  6027  f1stres  6050  f2ndres  6051  df1st2  6109  df2nd2  6110  dftpos2  6151  tfr2a  6211  freccllem  6292  frecfcllem  6294  frecsuclem  6296  djuf1olemr  6932  divfnzn  9406  cnmptid  12439  xmsxmet2  12621  msmet2  12622