Theorem ssres2 5990

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 5666	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × V) ⊆ (𝐵 × V))
2		sslin 4194	. . 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 5659	. 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V))
5		df-res 5659	. 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V))
6	3, 4, 5	3sstr4g 3989	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵))

Syntax hints:   → wi 4  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904   × cxp 5645   ↾ cres 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-in 3911  df-ss 3921  df-opab 5163  df-xp 5653  df-res 5659

This theorem is referenced by:  imass2  6091  1stcof  8000  2ndcof  8001  tfrlem15  8363  gsum2dlem2  20011  txkgen  23712  funpsstri  36116  eldisjss  39337  resnonrel  44168  mptrcllem  44189  rtrclexi  44197  cnvrcl0  44201  relexpss1d  44281  relexp0a  44292  supcnvlimsup  46314