Theorem ssres2 5085

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 4880	. . 3 |- ( A C_ B -> ( A X. _V ) C_ ( B X. _V ) )
2		sslin 3457	. . 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 4781	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
5		df-res 4781	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
6	3, 4, 5	3sstr4g 3291	1 |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )

Syntax hints:   -> wi 4   _Vcvv 2821   i^i cin 3219   C_ wss 3220   X. cxp 4767   |` cres 4771

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-opab 4188  df-xp 4775  df-res 4781

This theorem is referenced by:  imass2  5158  resasplitss  5564  fnsnsplitss  5905  1stcof  6387  2ndcof  6388