Theorem reseq1i 4887

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 4885	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1348   |` cres 4613

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 4623

This theorem is referenced by:  reseq12i  4889  resindm  4933  resmpt  4939  resmpt3  4940  resmptf  4941  opabresid  4944  rescnvcnv  5073  coires1  5128  fcoi1  5378  fvsnun1  5693  fvsnun2  5694  resoprab  5949  resmpo  5951  ofmres  6115  f1stres  6138  f2ndres  6139  df1st2  6198  df2nd2  6199  dftpos2  6240  tfr2a  6300  freccllem  6381  frecfcllem  6383  frecsuclem  6385  djuf1olemr  7031  divfnzn  9580  cnmptid  13075  xmsxmet2  13257  msmet2  13258