Theorem ssres2 4992

Description: Subclass theorem for restriction. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 22-Mar-1998.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
ssres2	⊢ (A ⊆ B → (C ↾ A) ⊆ (C ↾ B))

Proof of Theorem ssres2

Step	Hyp	Ref	Expression
1		xpss1 4857	. . 3 ⊢ (A ⊆ B → (A × V) ⊆ (B × V))
2		sslin 3482	. . 3 ⊢ ((A × V) ⊆ (B × V) → (C ∩ (A × V)) ⊆ (C ∩ (B × V)))
3	1, 2	syl 15	. 2 ⊢ (A ⊆ B → (C ∩ (A × V)) ⊆ (C ∩ (B × V)))
4		df-res 4789	. 2 ⊢ (C ↾ A) = (C ∩ (A × V))
5		df-res 4789	. 2 ⊢ (C ↾ B) = (C ∩ (B × V))
6	3, 4, 5	3sstr4g 3313	1 ⊢ (A ⊆ B → (C ↾ A) ⊆ (C ↾ B))

This proof depends on syntax axioms:   → wi 4  Vcvv 2860   ∩ cin 3209   ⊆ wss 3258   × cxp 4771   ↾ cres 4775

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 proof 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-xp 4785  df-res 4789

This theorem is used by:  imass2  5025