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Theorem ssres 6009
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 4232 . 2 (𝐴𝐵 → (𝐴 ∩ (𝐶 × V)) ⊆ (𝐵 ∩ (𝐶 × V)))
2 df-res 5690 . 2 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
3 df-res 5690 . 2 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
41, 2, 33sstr4g 4022 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  Vcvv 3461  cin 3943  wss 3944   × cxp 5676  cres 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-in 3951  df-ss 3961  df-res 5690
This theorem is referenced by:  imass1  6106  marypha1lem  9458  sspg  30610  ssps  30612  sspn  30618
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