Theorem reseq1i 4679

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 4677	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 7	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1287   |` cres 4415

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-res 4425

This theorem is referenced by:  reseq12i  4681  resindm  4723  resmpt  4729  resmpt3  4730  resmptf  4731  opabresid  4734  rescnvcnv  4861  coires1  4916  fcoi1  5156  fvsnun1  5459  fvsnun2  5460  resoprab  5700  resmpt2  5702  ofmres  5866  f1stres  5889  f2ndres  5890  df1st2  5943  df2nd2  5944  dftpos2  5982  tfr2a  6042  freccllem  6123  frecfcllem  6125  frecsuclem  6127  djuf1olemr  6693  divfnzn  9041