Theorem ssres 5344

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 3800	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 × V)) ⊆ (𝐵 ∩ (𝐶 × V)))
2		df-res 5050	. 2 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
3		df-res 5050	. 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
4	1, 2, 3	3sstr4g 3609	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶))

Syntax hints:   → wi 4  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   × cxp 5036   ↾ cres 5040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-res 5050

This theorem is referenced by:  imass1  5419  marypha1lem  8222  sspg  26967  ssps  26969  sspn  26975