Theorem reseq1 5007

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 3401	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4737	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4737	. 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 1397   _Vcvv 2802   i^i cin 3199   X. cxp 4723   |` cres 4727

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-res 4737

This theorem is referenced by:  reseq1i  5009  reseq1d  5012  imaeq1  5071  relcoi1  5268  tfr0dm  6487  tfrlemiex  6496  tfr1onlemex  6512  tfr1onlemaccex  6513  tfrcllemsucaccv  6519  tfrcllembxssdm  6521  tfrcllemex  6525  tfrcllemaccex  6526  tfrcllemres  6527  pmresg  6844  lmbr  14936