Theorem reseq1 4903

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 3331	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4640	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4640	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3eqtr4g 2235	1 |- ( A = B -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1353   _Vcvv 2739   i^i cin 3130   X. cxp 4626   |` cres 4630

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-res 4640

This theorem is referenced by:  reseq1i  4905  reseq1d  4908  imaeq1  4967  relcoi1  5162  tfr0dm  6325  tfrlemiex  6334  tfr1onlemex  6350  tfr1onlemaccex  6351  tfrcllemsucaccv  6357  tfrcllembxssdm  6359  tfrcllemex  6363  tfrcllemaccex  6364  tfrcllemres  6365  pmresg  6678  lmbr  13798