Theorem reseq1i 4885

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 4883	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1348   |` cres 4611

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-res 4621

This theorem is referenced by:  reseq12i  4887  resindm  4931  resmpt  4937  resmpt3  4938  resmptf  4939  opabresid  4942  rescnvcnv  5071  coires1  5126  fcoi1  5376  fvsnun1  5690  fvsnun2  5691  resoprab  5946  resmpo  5948  ofmres  6112  f1stres  6135  f2ndres  6136  df1st2  6195  df2nd2  6196  dftpos2  6237  tfr2a  6297  freccllem  6378  frecfcllem  6380  frecsuclem  6382  djuf1olemr  7027  divfnzn  9567  cnmptid  13034  xmsxmet2  13216  msmet2  13217