Theorem reseq1 4953

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 3367	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4687	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4687	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3eqtr4g 2263	1 |- ( A = B -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1373   _Vcvv 2772   i^i cin 3165   X. cxp 4673   |` cres 4677

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-res 4687

This theorem is referenced by:  reseq1i  4955  reseq1d  4958  imaeq1  5017  relcoi1  5214  tfr0dm  6408  tfrlemiex  6417  tfr1onlemex  6433  tfr1onlemaccex  6434  tfrcllemsucaccv  6440  tfrcllembxssdm  6442  tfrcllemex  6446  tfrcllemaccex  6447  tfrcllemres  6448  pmresg  6763  lmbr  14685