Theorem reseq1i 4903

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 4901	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1353   |` cres 4628

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 2739  df-in 3135  df-res 4638

This theorem is referenced by:  reseq12i  4905  resindm  4949  resmpt  4955  resmpt3  4956  resmptf  4957  opabresid  4960  rescnvcnv  5091  coires1  5146  fcoi1  5396  fvsnun1  5713  fvsnun2  5714  resoprab  5970  resmpo  5972  ofmres  6136  f1stres  6159  f2ndres  6160  df1st2  6219  df2nd2  6220  dftpos2  6261  tfr2a  6321  freccllem  6402  frecfcllem  6404  frecsuclem  6406  djuf1olemr  7052  divfnzn  9620  cnmptid  13751  xmsxmet2  13933  msmet2  13934