Theorem ssres2 4969

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 4769	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × V) ⊆ (𝐵 × V))
2		sslin 3385	. . 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 4671	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
5		df-res 4671	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
6	3, 4, 5	3sstr4g 3222	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4  Vcvv 2760   ∩ cin 3152   ⊆ wss 3153   × cxp 4657   ↾ cres 4661

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-opab 4091  df-xp 4665  df-res 4671

This theorem is referenced by:  imass2  5041  resasplitss  5433  fnsnsplitss  5757  1stcof  6216  2ndcof  6217