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Theorem trcoss2 36035
 Description: Equivalent expressions for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 4-Jul-2020.) (Revised by Peter Mazsa, 16-Oct-2021.)
Assertion
Ref Expression
trcoss2 (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
Distinct variable groups:   𝑦,𝑅   𝑥,𝑦   𝑦,𝑧
Allowed substitution hints:   𝑅(𝑥,𝑧)

Proof of Theorem trcoss2
StepHypRef Expression
1 alcom 2160 . . 3 (∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
21albii 1821 . 2 (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
3 19.23v 1943 . . . 4 (∀𝑦(𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅) ↔ (∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
4 eleccossin 36034 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧)))
54el2v 3449 . . . . . . 7 (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧))
65bicomi 227 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅))
7 brcoss3 35989 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥𝑅𝑧 ↔ ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
87el2v 3449 . . . . . 6 (𝑥𝑅𝑧 ↔ ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅)
96, 8imbi12i 354 . . . . 5 (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
109albii 1821 . . . 4 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦(𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
11 n0 4263 . . . . 5 (([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅))
1211imbi1i 353 . . . 4 ((([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅) ↔ (∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
133, 10, 123bitr4i 306 . . 3 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
14132albii 1822 . 2 (∀𝑥𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
152, 14bitri 278 1 (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  Vcvv 3442   ∩ cin 3882  ∅c0 4246   class class class wbr 5034  ◡ccnv 5522  [cec 8288   ≀ ccoss 35764 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 5171  ax-nul 5178  ax-pr 5299 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-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5035  df-opab 5097  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ec 8292  df-coss 35970 This theorem is referenced by:  eqvrelcoss4  36166
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