Theorem reseq1 4940

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 3357	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4675	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4675	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3eqtr4g 2254	1 |- ( A = B -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1364   _Vcvv 2763   i^i cin 3156   X. cxp 4661   |` cres 4665

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 4675

This theorem is referenced by:  reseq1i  4942  reseq1d  4945  imaeq1  5004  relcoi1  5201  tfr0dm  6380  tfrlemiex  6389  tfr1onlemex  6405  tfr1onlemaccex  6406  tfrcllemsucaccv  6412  tfrcllembxssdm  6414  tfrcllemex  6418  tfrcllemaccex  6419  tfrcllemres  6420  pmresg  6735  lmbr  14449