Theorem cossss 39164

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 5155	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦))
2		ssbr 5155	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑧 → 𝑥𝐵𝑧))
3	1, 2	anim12d 620	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → (𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)))
4	3	eximdv 1947	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)))
5	4	ssopab2dv 5536	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)})
6		df-coss 39150	. 2 ⊢ ≀ 𝐴 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)}
7		df-coss 39150	. 2 ⊢ ≀ 𝐵 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)}
8	5, 6, 7	3sstr4g 3990	1 ⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1809   ⊆ wss 3905   class class class wbr 5109  {copab 5173   ≀ ccoss 38832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ss 3922  df-br 5110  df-opab 5174  df-coss 39150

This theorem is referenced by:  funALTVss  39433