Theorem reseq1i 5059

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 5057	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

This proof depends on syntax axioms:   = wceq 1402   |` cres 4776

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-res 4786

This theorem is used by:  reseq12i  5061  resindm  5105  resmpt  5111  resmpt3  5112  resmptf  5113  opabresid  5116  rescnvcnv  5250  coires1  5305  fresaunres1disj  5571  fcoi1  5572  fvsnun1  5912  fvsnun2  5913  resoprab  6184  resmpo  6186  ofmres  6369  f1stres  6393  f2ndres  6394  df1st2  6455  df2nd2  6456  dftpos2  6532  tfr2a  6592  freccllem  6673  frecfcllem  6675  frecsuclem  6677  djuf1olemr  7394  divfnzn  10021  gsummptfidmadd  14161  cnmptid  15382  xmsxmet2  15564  msmet2  15565  cnfldms  15637