Theorem reseq1i 4974

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 4972	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1373   |` cres 4695

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-res 4705

This theorem is referenced by:  reseq12i  4976  resindm  5020  resmpt  5026  resmpt3  5027  resmptf  5028  opabresid  5031  rescnvcnv  5164  coires1  5219  fcoi1  5478  fvsnun1  5804  fvsnun2  5805  resoprab  6064  resmpo  6066  ofmres  6244  f1stres  6268  f2ndres  6269  df1st2  6328  df2nd2  6329  dftpos2  6370  tfr2a  6430  freccllem  6511  frecfcllem  6513  frecsuclem  6515  djuf1olemr  7182  divfnzn  9777  cnmptid  14868  xmsxmet2  15050  msmet2  15051  cnfldms  15123