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Theorem cossss 36144
 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 5080 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑦𝑥𝐵𝑦))
2 ssbr 5080 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑧𝑥𝐵𝑧))
31, 2anim12d 611 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑥𝐴𝑧) → (𝑥𝐵𝑦𝑥𝐵𝑧)))
43eximdv 1918 . . 3 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)))
54ssopab2dv 5412 . 2 (𝐴𝐵 → {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)})
6 df-coss 36133 . 2 𝐴 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧)}
7 df-coss 36133 . 2 𝐵 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)}
85, 6, 73sstr4g 3939 1 (𝐴𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781   ⊆ wss 3860   class class class wbr 5036  {copab 5098   ≀ ccoss 35927 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3867  df-ss 3877  df-br 5037  df-opab 5099  df-coss 36133 This theorem is referenced by:  funALTVss  36406
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