Theorem reseq1 4808

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 3265	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4546	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4546	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3eqtr4g 2195	1 |- ( A = B -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1331   _Vcvv 2681   i^i cin 3065   X. cxp 4532   |` cres 4536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-res 4546

This theorem is referenced by:  reseq1i  4810  reseq1d  4813  imaeq1  4871  relcoi1  5065  tfr0dm  6212  tfrlemiex  6221  tfr1onlemex  6237  tfr1onlemaccex  6238  tfrcllemsucaccv  6244  tfrcllembxssdm  6246  tfrcllemex  6250  tfrcllemaccex  6251  tfrcllemres  6252  pmresg  6563  lmbr  12371