Theorem reseq1i 5034

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 5032	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1398   |` cres 4751

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-res 4761

This theorem is referenced by:  reseq12i  5036  resindm  5080  resmpt  5086  resmpt3  5087  resmptf  5088  opabresid  5091  rescnvcnv  5225  coires1  5280  fresaunres1disj  5546  fcoi1  5547  fvsnun1  5881  fvsnun2  5882  resoprab  6149  resmpo  6151  ofmres  6329  f1stres  6353  f2ndres  6354  df1st2  6415  df2nd2  6416  dftpos2  6492  tfr2a  6552  freccllem  6633  frecfcllem  6635  frecsuclem  6637  djuf1olemr  7345  divfnzn  9953  cnmptid  15146  xmsxmet2  15328  msmet2  15329  cnfldms  15401