Theorem reseq1 4771

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 3236	. 2 |- ( A = B -> ( A i^i ( C X. _V ) ) = ( B i^i ( C X. _V ) ) )
2		df-res 4511	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4511	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3eqtr4g 2172	1 |- ( A = B -> ( A |` C ) = ( B |` C ) )

Syntax hints:   -> wi 4   = wceq 1314   _Vcvv 2657   i^i cin 3036   X. cxp 4497   |` cres 4501

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-in 3043  df-res 4511

This theorem is referenced by:  reseq1i  4773  reseq1d  4776  imaeq1  4834  relcoi1  5028  tfr0dm  6173  tfrlemiex  6182  tfr1onlemex  6198  tfr1onlemaccex  6199  tfrcllemsucaccv  6205  tfrcllembxssdm  6207  tfrcllemex  6211  tfrcllemaccex  6212  tfrcllemres  6213  pmresg  6524  lmbr  12224