Theorem reseq1i 4939

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 4937	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1364   |` cres 4662

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 3160  df-res 4672

This theorem is referenced by:  reseq12i  4941  resindm  4985  resmpt  4991  resmpt3  4992  resmptf  4993  opabresid  4996  rescnvcnv  5129  coires1  5184  fcoi1  5435  fvsnun1  5756  fvsnun2  5757  resoprab  6015  resmpo  6017  ofmres  6190  f1stres  6214  f2ndres  6215  df1st2  6274  df2nd2  6275  dftpos2  6316  tfr2a  6376  freccllem  6457  frecfcllem  6459  frecsuclem  6461  djuf1olemr  7115  divfnzn  9689  cnmptid  14460  xmsxmet2  14642  msmet2  14643  cnfldms  14715