Theorem reseq1i 5007

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 5005	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1395   |` cres 4725

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 2802  df-in 3204  df-res 4735

This theorem is referenced by:  reseq12i  5009  resindm  5053  resmpt  5059  resmpt3  5060  resmptf  5061  opabresid  5064  rescnvcnv  5197  coires1  5252  fcoi1  5514  fvsnun1  5846  fvsnun2  5847  resoprab  6112  resmpo  6114  ofmres  6293  f1stres  6317  f2ndres  6318  df1st2  6379  df2nd2  6380  dftpos2  6422  tfr2a  6482  freccllem  6563  frecfcllem  6565  frecsuclem  6567  djuf1olemr  7244  divfnzn  9845  cnmptid  14995  xmsxmet2  15177  msmet2  15178  cnfldms  15250