Theorem ssres 5962

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 4183	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 × V)) ⊆ (𝐵 ∩ (𝐶 × V)))
2		df-res 5636	. 2 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
3		df-res 5636	. 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
4	1, 2, 3	3sstr4g 3976	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶))

Syntax hints:   → wi 4  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890   × cxp 5622   ↾ cres 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-in 3897  df-ss 3907  df-res 5636

This theorem is referenced by:  imass1  6060  marypha1lem  9339  sspg  30814  ssps  30816  sspn  30822