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Theorem trcoss2 37988
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 2148 . . 3 (∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
21albii 1813 . 2 (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
3 19.23v 1937 . . . 4 (∀𝑦(𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅) ↔ (∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
4 eleccossin 37987 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧)))
54el2v 3481 . . . . . . 7 (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧))
65bicomi 223 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅))
7 brcoss3 37937 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥𝑅𝑧 ↔ ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
87el2v 3481 . . . . . 6 (𝑥𝑅𝑧 ↔ ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅)
96, 8imbi12i 349 . . . . 5 (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
109albii 1813 . . . 4 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦(𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
11 n0 4350 . . . . 5 (([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅))
1211imbi1i 348 . . . 4 ((([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅) ↔ (∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
133, 10, 123bitr4i 302 . . 3 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
14132albii 1814 . 2 (∀𝑥𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
152, 14bitri 274 1 (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531  wex 1773  wcel 2098  wne 2937  Vcvv 3473  cin 3948  c0 4326   class class class wbr 5152  ccnv 5681  [cec 8729  ccoss 37681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8733  df-coss 37915
This theorem is referenced by:  eqvrelcoss4  38124
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