Theorem ssres 4990

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

Assertion

Ref	Expression
ssres	⊢ (A ⊆ B → (A ↾ C) ⊆ (B ↾ C))

Proof of Theorem ssres

Step	Hyp	Ref	Expression
1		ssrin 3480	. 2 ⊢ (A ⊆ B → (A ∩ (C × V)) ⊆ (B ∩ (C × V)))
2		df-res 4788	. 2 ⊢ (A ↾ C) = (A ∩ (C × V))
3		df-res 4788	. 2 ⊢ (B ↾ C) = (B ∩ (C × V))
4	1, 2, 3	3sstr4g 3312	1 ⊢ (A ⊆ B → (A ↾ C) ⊆ (B ↾ C))

Syntax hints:   → wi 4  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257   × cxp 4770   ↾ cres 4774

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-res 4788

This theorem is referenced by:  imass1  5023