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Theorem trcoss2 35169
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 2095 . . 3 (∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
21albii 1782 . 2 (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
3 19.23v 1901 . . . 4 (∀𝑦(𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅) ↔ (∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
4 eleccossin 35168 . . . . . . . 8 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧)))
54el2v 3416 . . . . . . 7 (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ↔ (𝑥𝑅𝑦𝑦𝑅𝑧))
65bicomi 216 . . . . . 6 ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅))
7 brcoss3 35123 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥𝑅𝑧 ↔ ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
87el2v 3416 . . . . . 6 (𝑥𝑅𝑧 ↔ ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅)
96, 8imbi12i 343 . . . . 5 (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
109albii 1782 . . . 4 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦(𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
11 n0 4190 . . . . 5 (([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅))
1211imbi1i 342 . . . 4 ((([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅) ↔ (∃𝑦 𝑦 ∈ ([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
133, 10, 123bitr4i 295 . . 3 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
14132albii 1783 . 2 (∀𝑥𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
152, 14bitri 267 1 (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑥𝑧(([𝑥] ≀ 𝑅 ∩ [𝑧] ≀ 𝑅) ≠ ∅ → ([𝑥]𝑅 ∩ [𝑧]𝑅) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wal 1505  wex 1742  wcel 2050  wne 2961  Vcvv 3409  cin 3822  c0 4172   class class class wbr 4923  ccnv 5400  [cec 8081  ccoss 34897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-xp 5407  df-rel 5408  df-cnv 5409  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-ec 8085  df-coss 35104
This theorem is referenced by:  eqvrelcoss4  35300
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