Theorem reseq1 4663

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 3178	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4411	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4411	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3eqtr4g 2140	1 |- ( A = B -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1285   _Vcvv 2612   i^i cin 2983   X. cxp 4397   |` cres 4401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-res 4411

This theorem is referenced by:  reseq1i  4665  reseq1d  4668  imaeq1  4722  relcoi1  4914  tfr0dm  6017  tfrlemiex  6026  tfr1onlemex  6042  tfr1onlemaccex  6043  tfrcllemsucaccv  6049  tfrcllembxssdm  6051  tfrcllemex  6055  tfrcllemaccex  6056  tfrcllemres  6057  pmresg  6361