Theorem reseq1i 4905

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 4903	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1353   |` cres 4630

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-res 4640

This theorem is referenced by:  reseq12i  4907  resindm  4951  resmpt  4957  resmpt3  4958  resmptf  4959  opabresid  4962  rescnvcnv  5093  coires1  5148  fcoi1  5398  fvsnun1  5715  fvsnun2  5716  resoprab  5973  resmpo  5975  ofmres  6139  f1stres  6162  f2ndres  6163  df1st2  6222  df2nd2  6223  dftpos2  6264  tfr2a  6324  freccllem  6405  frecfcllem  6407  frecsuclem  6409  djuf1olemr  7055  divfnzn  9623  cnmptid  13820  xmsxmet2  14002  msmet2  14003