Theorem cocossss 35841

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 35828	. . 3 ⊢ Rel ≀ ≀ 𝑅
2		ssrel3 35716	. . 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 35836	. . . . . . . . 9 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑦 ≀ 𝑅𝑥 ∧ 𝑦 ≀ 𝑅𝑧)))
5	4	el2v 3448	. . . . . . . 8 ⊢ (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑦 ≀ 𝑅𝑥 ∧ 𝑦 ≀ 𝑅𝑧))
6		brcosscnvcoss 35839	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦))
7	6	el2v 3448	. . . . . . . . . 10 ⊢ (𝑦 ≀ 𝑅𝑥 ↔ 𝑥 ≀ 𝑅𝑦)
8	7	anbi1i 626	. . . . . . . . 9 ⊢ ((𝑦 ≀ 𝑅𝑥 ∧ 𝑦 ≀ 𝑅𝑧) ↔ (𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧))
9	8	exbii 1849	. . . . . . . 8 ⊢ (∃𝑦(𝑦 ≀ 𝑅𝑥 ∧ 𝑦 ≀ 𝑅𝑧) ↔ ∃𝑦(𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧))
10	5, 9	bitri 278	. . . . . . 7 ⊢ (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧))
11	10	imbi1i 353	. . . . . 6 ⊢ ((𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ (∃𝑦(𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
12		19.23v 1943	. . . . . 6 ⊢ (∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧) ↔ (∃𝑦(𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
13	11, 12	bitr4i 281	. . . . 5 ⊢ ((𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ ∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
14	13	albii 1821	. . . 4 ⊢ (∀𝑧(𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ ∀𝑧∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
15		alcom 2160	. . . 4 ⊢ (∀𝑧∀𝑦((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧) ↔ ∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
16	14, 15	bitri 278	. . 3 ⊢ (∀𝑧(𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ ∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
17	16	albii 1821	. 2 ⊢ (∀𝑥∀𝑧(𝑥 ≀ ≀ 𝑅𝑧 → 𝑥𝑆𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))
18	3, 17	bitri 278	1 ⊢ ( ≀ ≀ 𝑅 ⊆ 𝑆 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥𝑆𝑧))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030  Rel wrel 5524   ≀ ccoss 35613

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-coss 35819

This theorem is referenced by:  eqvrelcoss2  36014