Theorem cocossss 38985

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 38972	. . 3 ⊢ Rel ≀ ≀ 𝑅
2		ssrel3 5754	. . 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 38980	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑦 ≀ 𝑅𝑥 ∧ 𝑦 ≀ 𝑅𝑧)))
5	4	el2v 3460	. . . . . . . 8 ⊢ (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑦 ≀ 𝑅𝑥 ∧ 𝑦 ≀ 𝑅𝑧))
6		brcosscnvcoss 38983	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦))
7	6	el2v 3460	. . . . . . . . . 10 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)
8	7	anbi1i 633	. . . . . . . . 9 ⊢ ((𝑦 ≀ 𝑅𝑥 ∧ 𝑦 ≀ 𝑅𝑧) ↔ (𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧))
9	8	exbii 1867	. . . . . . . 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 351	. . . . . 6 ⊢ ((𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ (∃𝑦(𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
12		19.23v 1961	. . . . . 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 1838	. . . 4 ⊢ (∀𝑧(𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ ∀𝑧∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
15		alcom 2192	. . . 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 1838	. 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 399  ∀wal 1557  ∃wex 1798  Vcvv 3453   ⊆ wss 3902   class class class wbr 5097  Rel wrel 5648   ≀ ccoss 38642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-11 2190  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649  df-rel 5650  df-coss 38960

This theorem is referenced by:  eqvrelcoss2  39162