Theorem ssres 4972

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 3388	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 × V)) ⊆ (𝐵 ∩ (𝐶 × V)))
2		df-res 4675	. 2 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
3		df-res 4675	. 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
4	1, 2, 3	3sstr4g 3226	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ↾ 𝐶) ⊆ (𝐵 ↾ 𝐶))

Syntax hints:   → wi 4  Vcvv 2763   ∩ cin 3156   ⊆ wss 3157   × cxp 4661   ↾ cres 4665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-res 4675

This theorem is referenced by:  imass1  5044