Theorem cossss 38766

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 5144	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦))
2		ssbr 5144	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑧 → 𝑥𝐵𝑧))
3	1, 2	anim12d 610	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → (𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)))
4	3	eximdv 1919	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)))
5	4	ssopab2dv 5507	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)})
6		df-coss 38752	. 2 ⊢ ≀ 𝐴 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)}
7		df-coss 38752	. 2 ⊢ ≀ 𝐵 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)}
8	5, 6, 7	3sstr4g 3989	1 ⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1781   ⊆ wss 3903   class class class wbr 5100  {copab 5162   ≀ ccoss 38434

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ss 3920  df-br 5101  df-opab 5163  df-coss 38752

This theorem is referenced by:  funALTVss  39035