Theorem ssres2 5964

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 5650	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × V) ⊆ (𝐵 × V))
2		sslin 4202	. . 3 ⊢ ((𝐴 × V) ⊆ (𝐵 × V) → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V)))
3	1, 2	syl 17	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V)))
4		df-res 5643	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
5		df-res 5643	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
6	3, 4, 5	3sstr4g 3997	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911   × cxp 5629   ↾ cres 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-in 3918  df-ss 3928  df-opab 5165  df-xp 5637  df-res 5643

This theorem is referenced by:  imass2  6062  1stcof  7977  2ndcof  7978  tfrlem15  8337  gsum2dlem2  19877  txkgen  23515  funpsstri  35726  eldisjss  38703  resnonrel  43554  mptrcllem  43575  rtrclexi  43583  cnvrcl0  43587  relexpss1d  43667  relexp0a  43678  supcnvlimsup  45711