Theorem reseq1 4999

Description: Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)

Assertion

Ref	Expression
reseq1	|- ( A = B -> ( A |` C ) = ( B |` C ) )

Proof of Theorem reseq1

Step	Hyp	Ref	Expression
1		ineq1 3398	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4731	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4731	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3eqtr4g 2287	1 |- ( A = B -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1395   _Vcvv 2799   i^i cin 3196   X. cxp 4717   |` cres 4721

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-res 4731

This theorem is referenced by:  reseq1i  5001  reseq1d  5004  imaeq1  5063  relcoi1  5260  tfr0dm  6474  tfrlemiex  6483  tfr1onlemex  6499  tfr1onlemaccex  6500  tfrcllemsucaccv  6506  tfrcllembxssdm  6508  tfrcllemex  6512  tfrcllemaccex  6513  tfrcllemres  6514  pmresg  6831  lmbr  14902