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Theorem cossss 38407
Description: Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.)
Assertion
Ref Expression
cossss (𝐴𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)

Proof of Theorem cossss
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssbr 5192 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑦𝑥𝐵𝑦))
2 ssbr 5192 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑧𝑥𝐵𝑧))
31, 2anim12d 609 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑥𝐴𝑧) → (𝑥𝐵𝑦𝑥𝐵𝑧)))
43eximdv 1915 . . 3 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)))
54ssopab2dv 5561 . 2 (𝐴𝐵 → {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)})
6 df-coss 38393 . 2 𝐴 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧)}
7 df-coss 38393 . 2 𝐵 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)}
85, 6, 73sstr4g 4041 1 (𝐴𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1776  wss 3963   class class class wbr 5148  {copab 5210  ccoss 38162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ss 3980  df-br 5149  df-opab 5211  df-coss 38393
This theorem is referenced by:  funALTVss  38681
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