Theorem ssres2 5065

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 4860	. . 3 |- ( A C_ B -> ( A X. _V ) C_ ( B X. _V ) )
2		sslin 3447	. . 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 14	. 2 |- ( A C_ B -> ( C i^i ( A X. _V ) ) C_ ( C i^i ( B X. _V ) ) )
4		df-res 4761	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
5		df-res 4761	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
6	3, 4, 5	3sstr4g 3281	1 |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )

Syntax hints:   -> wi 4   _Vcvv 2813   i^i cin 3210   C_ wss 3211   X. cxp 4747   |` cres 4751

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-opab 4172  df-xp 4755  df-res 4761

This theorem is referenced by:  imass2  5138  resasplitss  5544  fnsnsplitss  5883  1stcof  6357  2ndcof  6358