Theorem ssres 6008

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

Assertion

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

Proof of Theorem ssres

Step	Hyp	Ref	Expression
1		ssrin 4233	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 × V)) ⊆ (𝐵 ∩ (𝐶 × V)))
2		df-res 5688	. 2 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
3		df-res 5688	. 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
4	1, 2, 3	3sstr4g 4027	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶))

Syntax hints:   → wi 4  Vcvv 3473   ∩ cin 3947   ⊆ wss 3948   × cxp 5674   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965  df-res 5688

This theorem is referenced by:  imass1  6100  marypha1lem  9434  sspg  30263  ssps  30265  sspn  30271