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Theorem clsslem 14878
Description: The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020.)
Assertion
Ref Expression
clsslem (𝑅𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} ⊆ {𝑟 ∣ (𝑆𝑟𝜑)})
Distinct variable groups:   𝑅,𝑟   𝑆,𝑟
Allowed substitution hint:   𝜑(𝑟)

Proof of Theorem clsslem
StepHypRef Expression
1 sstr2 3955 . . . 4 (𝑅𝑆 → (𝑆𝑟𝑅𝑟))
21anim1d 612 . . 3 (𝑅𝑆 → ((𝑆𝑟𝜑) → (𝑅𝑟𝜑)))
32ss2abdv 4024 . 2 (𝑅𝑆 → {𝑟 ∣ (𝑆𝑟𝜑)} ⊆ {𝑟 ∣ (𝑅𝑟𝜑)})
4 intss 4934 . 2 ({𝑟 ∣ (𝑆𝑟𝜑)} ⊆ {𝑟 ∣ (𝑅𝑟𝜑)} → {𝑟 ∣ (𝑅𝑟𝜑)} ⊆ {𝑟 ∣ (𝑆𝑟𝜑)})
53, 4syl 17 1 (𝑅𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} ⊆ {𝑟 ∣ (𝑆𝑟𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  {cab 2710  wss 3914   cint 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3449  df-in 3921  df-ss 3931  df-int 4912
This theorem is referenced by:  trclsslem  14884
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