Theorem reseq1i 4943

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 4941	. 2 |- ( A = B -> ( A |` C ) = ( B |` C ) )
3	1, 2	ax-mp 5	1 |- ( A |` C ) = ( B |` C )

Syntax hints:   = wceq 1364   |` cres 4666

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-res 4676

This theorem is referenced by:  reseq12i  4945  resindm  4989  resmpt  4995  resmpt3  4996  resmptf  4997  opabresid  5000  rescnvcnv  5133  coires1  5188  fcoi1  5441  fvsnun1  5762  fvsnun2  5763  resoprab  6022  resmpo  6024  ofmres  6202  f1stres  6226  f2ndres  6227  df1st2  6286  df2nd2  6287  dftpos2  6328  tfr2a  6388  freccllem  6469  frecfcllem  6471  frecsuclem  6473  djuf1olemr  7129  divfnzn  9712  cnmptid  14601  xmsxmet2  14783  msmet2  14784  cnfldms  14856