Theorem ssres2 4656

Description: Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

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

Proof of Theorem ssres2

Step	Hyp	Ref	Expression
1		xpss1 4466	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × V) ⊆ (𝐵 × V))
2		sslin 3192	. . 3 ⊢ ((𝐴 × V) ⊆ (𝐵 × V) → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V)))
3	1, 2	syl 14	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V)))
4		df-res 4375	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
5		df-res 4375	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
6	3, 4, 5	3sstr4g 3040	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4  Vcvv 2601   ∩ cin 2972   ⊆ wss 2973   × cxp 4361   ↾ cres 4365

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-opab 3840  df-xp 4369  df-res 4375

This theorem is referenced by:  imass2  4721  resasplitss  5089  1stcof  5810  2ndcof  5811