Theorem ssres2 6007

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 5691	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × V) ⊆ (𝐵 × V))
2		sslin 4230	. . 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 5684	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
5		df-res 5684	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
6	3, 4, 5	3sstr4g 4023	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4  Vcvv 3470   ∩ cin 3944   ⊆ wss 3945   × cxp 5670   ↾ cres 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-in 3952  df-ss 3962  df-opab 5205  df-xp 5678  df-res 5684

This theorem is referenced by:  imass2  6100  1stcof  8017  2ndcof  8018  tfrlem15  8406  gsum2dlem2  19919  txkgen  23549  funpsstri  35355  eldisjss  38204  resnonrel  43016  mptrcllem  43037  rtrclexi  43045  cnvrcl0  43049  relexpss1d  43129  relexp0a  43140  supcnvlimsup  45122