Theorem ssres2 5131

Description: Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
ssres2	|- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )

Proof of Theorem ssres2

Step	Hyp	Ref	Expression
1		xpss1 4943	. . 3 |- ( A C_ B -> ( A X. _V ) C_ ( B X. _V ) )
2		sslin 3658	. . 3 |- ( ( A X. _V ) C_ ( B X. _V ) -> ( C i^i ( A X. _V ) ) C_ ( C i^i ( B X. _V ) ) )
3	1, 2	syl 17	. 2 |- ( A C_ B -> ( C i^i ( A X. _V ) ) C_ ( C i^i ( B X. _V ) ) )
4		df-res 4846	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
5		df-res 4846	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
6	3, 4, 5	3sstr4g 3473	1 |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )

Syntax hints:   -> wi 4   _Vcvv 3045   i^i cin 3403   C_ wss 3404   X. cxp 4832   |` cres 4836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-in 3411  df-ss 3418  df-opab 4462  df-xp 4840  df-res 4846

This theorem is referenced by:  imass2  5204  1stcof  6821  2ndcof  6822  tfrlem15  7110  gsum2dlem2  17603  txkgen  20667  funpsstri  30406  resnonrel  36198  mptrcllem  36220  rtrclexi  36228  cnvrcl0  36232  relexpss1d  36297  relexp0a  36308