Theorem reseq1 5013

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 3403	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4743	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4743	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3eqtr4g 2289	1 |- ( A = B -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1398   _Vcvv 2803   i^i cin 3200   X. cxp 4729   |` cres 4733

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-res 4743

This theorem is referenced by:  reseq1i  5015  reseq1d  5018  imaeq1  5077  relcoi1  5275  tfr0dm  6531  tfrlemiex  6540  tfr1onlemex  6556  tfr1onlemaccex  6557  tfrcllemsucaccv  6563  tfrcllembxssdm  6565  tfrcllemex  6569  tfrcllemaccex  6570  tfrcllemres  6571  pmresg  6888  lmbr  15007