Theorem ssres 4665

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 3198	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 × V)) ⊆ (𝐵 ∩ (𝐶 × V)))
2		df-res 4383	. 2 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
3		df-res 4383	. 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
4	1, 2, 3	3sstr4g 3041	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶))

Syntax hints:   → wi 4  Vcvv 2602   ∩ cin 2973   ⊆ wss 2974   × cxp 4369   ↾ cres 4373

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-res 4383

This theorem is referenced by:  imass1  4730