Theorem ssres 5045

Description: Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)

Assertion

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

Proof of Theorem ssres

Step	Hyp	Ref	Expression
1		ssrin 3434	. 2 |- ( A C_ B -> ( A i^i ( C X. _V ) ) C_ ( B i^i ( C X. _V ) ) )
2		df-res 4743	. 2 |- ( A |` C ) = ( A i^i ( C X. _V ) )
3		df-res 4743	. 2 |- ( B |` C ) = ( B i^i ( C X. _V ) )
4	1, 2, 3	3sstr4g 3271	1 |- ( A C_ B -> ( A |` C ) C_ ( B |` C ) )

Syntax hints:   -> wi 4   _Vcvv 2803   i^i cin 3200   C_ wss 3201   X. cxp 4729   |` cres 4733

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-res 4743

This theorem is referenced by:  imass1  5118