Theorem reseq1i 4955

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 4953	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1373   |` cres 4677

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-res 4687

This theorem is referenced by:  reseq12i  4957  resindm  5001  resmpt  5007  resmpt3  5008  resmptf  5009  opabresid  5012  rescnvcnv  5145  coires1  5200  fcoi1  5456  fvsnun1  5781  fvsnun2  5782  resoprab  6041  resmpo  6043  ofmres  6221  f1stres  6245  f2ndres  6246  df1st2  6305  df2nd2  6306  dftpos2  6347  tfr2a  6407  freccllem  6488  frecfcllem  6490  frecsuclem  6492  djuf1olemr  7156  divfnzn  9742  cnmptid  14753  xmsxmet2  14935  msmet2  14936  cnfldms  15008