Theorem ssres2 4933

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 4735	. . 3 |- ( A C_ B -> ( A X. _V ) C_ ( B X. _V ) )
2		sslin 3361	. . 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 4637	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
5		df-res 4637	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
6	3, 4, 5	3sstr4g 3198	1 |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )

Syntax hints:   -> wi 4   _Vcvv 2737   i^i cin 3128   C_ wss 3129   X. cxp 4623   |` cres 4627

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 2739  df-in 3135  df-ss 3142  df-opab 4064  df-xp 4631  df-res 4637

This theorem is referenced by:  imass2  5003  resasplitss  5394  fnsnsplitss  5714  1stcof  6161  2ndcof  6162