Theorem reseq1i 5054

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 5052	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1402   |` cres 4771

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 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 theorem 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 4781

This theorem is referenced by:  reseq12i  5056  resindm  5100  resmpt  5106  resmpt3  5107  resmptf  5108  opabresid  5111  rescnvcnv  5245  coires1  5300  fresaunres1disj  5566  fcoi1  5567  fvsnun1  5903  fvsnun2  5904  resoprab  6174  resmpo  6176  ofmres  6359  f1stres  6383  f2ndres  6384  df1st2  6445  df2nd2  6446  dftpos2  6522  tfr2a  6582  freccllem  6663  frecfcllem  6665  frecsuclem  6667  djuf1olemr  7384  divfnzn  10000  gsummptfidmadd  14138  cnmptid  15305  xmsxmet2  15487  msmet2  15488  cnfldms  15560