Theorem ssres2 4638

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 4448	. . 3 |- ( A C_ B -> ( A X. _V ) C_ ( B X. _V ) )
2		sslin 3163	. . 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 4357	. 2 |- ( C |` A ) = ( C i^i ( A X. _V ) )
5		df-res 4357	. 2 |- ( C |` B ) = ( C i^i ( B X. _V ) )
6	3, 4, 5	3sstr4g 2986	1 |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )

Syntax hints:   -> wi 4   _Vcvv 2557   i^i cin 2916   C_ wss 2917   X. cxp 4343   |` cres 4347

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-opab 3819  df-xp 4351  df-res 4357

This theorem is referenced by:  imass2  4701  resasplitss  5069  1stcof  5790  2ndcof  5791