Theorem reseq1i 5000

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 4998	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1395   |` cres 4720

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-res 4730

This theorem is referenced by:  reseq12i  5002  resindm  5046  resmpt  5052  resmpt3  5053  resmptf  5054  opabresid  5057  rescnvcnv  5190  coires1  5245  fcoi1  5505  fvsnun1  5835  fvsnun2  5836  resoprab  6099  resmpo  6101  ofmres  6279  f1stres  6303  f2ndres  6304  df1st2  6363  df2nd2  6364  dftpos2  6405  tfr2a  6465  freccllem  6546  frecfcllem  6548  frecsuclem  6550  djuf1olemr  7217  divfnzn  9812  cnmptid  14949  xmsxmet2  15131  msmet2  15132  cnfldms  15204