Theorem reseq1i 4938

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 4936	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1364   |` cres 4661

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-res 4671

This theorem is referenced by:  reseq12i  4940  resindm  4984  resmpt  4990  resmpt3  4991  resmptf  4992  opabresid  4995  rescnvcnv  5128  coires1  5183  fcoi1  5434  fvsnun1  5755  fvsnun2  5756  resoprab  6014  resmpo  6016  ofmres  6188  f1stres  6212  f2ndres  6213  df1st2  6272  df2nd2  6273  dftpos2  6314  tfr2a  6374  freccllem  6455  frecfcllem  6457  frecsuclem  6459  djuf1olemr  7113  divfnzn  9686  cnmptid  14449  xmsxmet2  14631  msmet2  14632