Theorem cocossss 35685

Description: Two ways of saying that cosets by cosets by 𝑅 is a subclass. (Contributed by Peter Mazsa, 17-Sep-2021.)

Assertion

Ref	Expression
cocossss	⊢ ( ≀ ≀ 𝑅 ⊆ 𝑆 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))

Distinct variable groups:   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem cocossss

Step	Hyp	Ref	Expression
1		relcoss 35672	. . 3 ⊢ Rel ≀ ≀ 𝑅
2		ssrel3 35560	. . 3 ⊢ (Rel ≀ ≀ 𝑅 → ( ≀ ≀ 𝑅 ⊆ 𝑆 ↔ ∀𝑥∀𝑧(𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧)))
3	1, 2	ax-mp 5	. 2 ⊢ ( ≀ ≀ 𝑅 ⊆ 𝑆 ↔ ∀𝑥∀𝑧(𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧))
4		brcoss 35680	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑦 ≀ 𝑅𝑥 ∧ 𝑦 ≀ 𝑅𝑧)))
5	4	el2v 3504	. . . . . . . 8 ⊢ (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑦 ≀ 𝑅𝑥 ∧ 𝑦 ≀ 𝑅𝑧))
6		brcosscnvcoss 35683	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦))
7	6	el2v 3504	. . . . . . . . . 10 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)
8	7	anbi1i 625	. . . . . . . . 9 ⊢ ((𝑦 ≀ 𝑅𝑥 ∧ 𝑦 ≀ 𝑅𝑧) ↔ (𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧))
9	8	exbii 1847	. . . . . . . 8 ⊢ (∃𝑦(𝑦 ≀ 𝑅𝑥 ∧ 𝑦 ≀ 𝑅𝑧) ↔ ∃𝑦(𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧))
10	5, 9	bitri 277	. . . . . . 7 ⊢ (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧))
11	10	imbi1i 352	. . . . . 6 ⊢ ((𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ (∃𝑦(𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
12		19.23v 1942	. . . . . 6 ⊢ (∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧) ↔ (∃𝑦(𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
13	11, 12	bitr4i 280	. . . . 5 ⊢ ((𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ ∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
14	13	albii 1819	. . . 4 ⊢ (∀𝑧(𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ ∀𝑧∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
15		alcom 2162	. . . 4 ⊢ (∀𝑧∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧) ↔ ∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
16	14, 15	bitri 277	. . 3 ⊢ (∀𝑧(𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ ∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
17	16	albii 1819	. 2 ⊢ (∀𝑥∀𝑧(𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
18	3, 17	bitri 277	1 ⊢ ( ≀ ≀ 𝑅 ⊆ 𝑆 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534  ∃wex 1779  Vcvv 3497   ⊆ wss 3939   class class class wbr 5069  Rel wrel 5563   ≀ ccoss 35457

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-coss 35663

This theorem is referenced by:  eqvrelcoss2  35858