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Theorem cossss 37233
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 5191 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑦𝑥𝐵𝑦))
2 ssbr 5191 . . . . 5 (𝐴𝐵 → (𝑥𝐴𝑧𝑥𝐵𝑧))
31, 2anim12d 610 . . . 4 (𝐴𝐵 → ((𝑥𝐴𝑦𝑥𝐴𝑧) → (𝑥𝐵𝑦𝑥𝐵𝑧)))
43eximdv 1921 . . 3 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)))
54ssopab2dv 5550 . 2 (𝐴𝐵 → {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)})
6 df-coss 37219 . 2 𝐴 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧)}
7 df-coss 37219 . 2 𝐵 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦𝑥𝐵𝑧)}
85, 6, 73sstr4g 4026 1 (𝐴𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wex 1782  wss 3947   class class class wbr 5147  {copab 5209  ccoss 36981
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-v 3477  df-in 3954  df-ss 3964  df-br 5148  df-opab 5210  df-coss 37219
This theorem is referenced by:  funALTVss  37507
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