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Theorem cocossss 34533
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 34520 . . 3 Rel ≀ ≀ 𝑅
2 ssrel3 34410 . . 3 (Rel ≀ ≀ 𝑅 → ( ≀ ≀ 𝑅𝑆 ↔ ∀𝑥𝑧(𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧)))
31, 2ax-mp 5 . 2 ( ≀ ≀ 𝑅𝑆 ↔ ∀𝑥𝑧(𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧))
4 brcoss 34528 . . . . . . . . 9 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑦𝑅𝑥𝑦𝑅𝑧)))
54el2v 34329 . . . . . . . 8 (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑦𝑅𝑥𝑦𝑅𝑧))
6 brcosscnvcoss 34531 . . . . . . . . . . 11 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦𝑅𝑥𝑥𝑅𝑦))
76el2v 34329 . . . . . . . . . 10 (𝑦𝑅𝑥𝑥𝑅𝑦)
87anbi1i 610 . . . . . . . . 9 ((𝑦𝑅𝑥𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧))
98exbii 1924 . . . . . . . 8 (∃𝑦(𝑦𝑅𝑥𝑦𝑅𝑧) ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧))
105, 9bitri 264 . . . . . . 7 (𝑥 ≀ ≀ 𝑅𝑧 ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧))
1110imbi1i 338 . . . . . 6 ((𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧) ↔ (∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
12 19.23v 2023 . . . . . 6 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧) ↔ (∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
1311, 12bitr4i 267 . . . . 5 ((𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧) ↔ ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
1413albii 1895 . . . 4 (∀𝑧(𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧) ↔ ∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
15 alcom 2193 . . . 4 (∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
1614, 15bitri 264 . . 3 (∀𝑧(𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
1716albii 1895 . 2 (∀𝑥𝑧(𝑥 ≀ ≀ 𝑅𝑧𝑥𝑆𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
183, 17bitri 264 1 ( ≀ ≀ 𝑅𝑆 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑆𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629  wex 1852  Vcvv 3351  wss 3723   class class class wbr 4786  Rel wrel 5254  ccoss 34315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-coss 34511
This theorem is referenced by: (None)
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