Theorem cossss 38426

Description: Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.)

Assertion

Ref	Expression
cossss	⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)

Proof of Theorem cossss

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssbr 5187	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦))
2		ssbr 5187	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑧 → 𝑥𝐵𝑧))
3	1, 2	anim12d 609	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → (𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)))
4	3	eximdv 1917	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)))
5	4	ssopab2dv 5556	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)})
6		df-coss 38412	. 2 ⊢ ≀ 𝐴 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)}
7		df-coss 38412	. 2 ⊢ ≀ 𝐵 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)}
8	5, 6, 7	3sstr4g 4037	1 ⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ⊆ wss 3951   class class class wbr 5143  {copab 5205   ≀ ccoss 38182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ss 3968  df-br 5144  df-opab 5206  df-coss 38412

This theorem is referenced by:  funALTVss  38700