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Theorem ssres 6023
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 4249 . 2 (𝐴𝐵 → (𝐴 ∩ (𝐶 × V)) ⊆ (𝐵 ∩ (𝐶 × V)))
2 df-res 5700 . 2 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
3 df-res 5700 . 2 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
41, 2, 33sstr4g 4040 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  Vcvv 3477  cin 3961  wss 3962   × cxp 5686  cres 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-in 3969  df-ss 3979  df-res 5700
This theorem is referenced by:  imass1  6121  marypha1lem  9470  sspg  30756  ssps  30758  sspn  30764
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