Theorem reseq1i 4608

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 4606	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 7	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1243   |` cres 4347

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-res 4357

This theorem is referenced by:  reseq12i  4610  resmpt  4656  resmpt3  4657  opabresid  4659  rescnvcnv  4783  coires1  4838  fcoi1  5070  fvsnun1  5360  fvsnun2  5361  resoprab  5597  resmpt2  5599  ofmres  5763  f1stres  5786  f2ndres  5787  df1st2  5840  df2nd2  5841  dftpos2  5876  tfr2a  5936  frecsuclem1  5987  frecsuclem2  5989  divfnzn  8556