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Theorem ssres 6021
Description: Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
ssres (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem ssres
StepHypRef Expression
1 ssrin 4242 . 2 (𝐴𝐵 → (𝐴 ∩ (𝐶 × V)) ⊆ (𝐵 ∩ (𝐶 × V)))
2 df-res 5697 . 2 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
3 df-res 5697 . 2 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
41, 2, 33sstr4g 4037 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  Vcvv 3480  cin 3950  wss 3951   × cxp 5683  cres 5687
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-ss 3968  df-res 5697
This theorem is referenced by:  imass1  6119  marypha1lem  9473  sspg  30747  ssps  30749  sspn  30755
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