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Theorem cocossss 35800
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 ( ≀ ≀ 𝑅𝑆 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
Distinct variable groups:   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem cocossss
StepHypRef Expression
1 relcoss 35787 . . 3 Rel ≀ ≀ 𝑅
2 ssrel3 35675 . . 3 (Rel ≀ ≀ 𝑅 → ( ≀ ≀ 𝑅𝑆 ↔ ∀𝑥𝑧(𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧)))
31, 2ax-mp 5 . 2 ( ≀ ≀ 𝑅𝑆 ↔ ∀𝑥𝑧(𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧))
4 brcoss 35795 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑦𝑅𝑥𝑦𝑅𝑧)))
54el2v 3476 . . . . . . . 8 (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑦𝑅𝑥𝑦𝑅𝑧))
6 brcosscnvcoss 35798 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
76el2v 3476 . . . . . . . . . 10 (𝑦𝑅𝑥𝑥𝑅𝑦)
87anbi1i 626 . . . . . . . . 9 ((𝑦𝑅𝑥𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧))
98exbii 1849 . . . . . . . 8 (∃𝑦(𝑦𝑅𝑥𝑦𝑅𝑧) ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧))
105, 9bitri 278 . . . . . . 7 (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧))
1110imbi1i 353 . . . . . 6 ((𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧) ↔ (∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
12 19.23v 1943 . . . . . 6 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧) ↔ (∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
1311, 12bitr4i 281 . . . . 5 ((𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧) ↔ ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
1413albii 1821 . . . 4 (∀𝑧(𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧) ↔ ∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
15 alcom 2163 . . . 4 (∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
1614, 15bitri 278 . . 3 (∀𝑧(𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
1716albii 1821 . 2 (∀𝑥𝑧(𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
183, 17bitri 278 1 ( ≀ ≀ 𝑅𝑆 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781  Vcvv 3469  wss 3908   class class class wbr 5042  Rel wrel 5537  ccoss 35572
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-coss 35778
This theorem is referenced by:  eqvrelcoss2  35973
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