Theorem reseq1 4972

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 3375	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4705	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4705	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3eqtr4g 2265	1 |- ( A = B -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1373   _Vcvv 2776   i^i cin 3173   X. cxp 4691   |` cres 4695

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-res 4705

This theorem is referenced by:  reseq1i  4974  reseq1d  4977  imaeq1  5036  relcoi1  5233  tfr0dm  6431  tfrlemiex  6440  tfr1onlemex  6456  tfr1onlemaccex  6457  tfrcllemsucaccv  6463  tfrcllembxssdm  6465  tfrcllemex  6469  tfrcllemaccex  6470  tfrcllemres  6471  pmresg  6786  lmbr  14800