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Theorem trcoss 35602
Description: Sufficient condition for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 26-Dec-2018.)
Assertion
Ref Expression
trcoss (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Distinct variable groups:   𝑢,𝑅,𝑥   𝑧,𝑅,𝑢   𝑦,𝑢,𝑥   𝑦,𝑧
Allowed substitution hint:   𝑅(𝑦)

Proof of Theorem trcoss
StepHypRef Expression
1 moantr 35497 . . . . 5 (∃*𝑢 𝑢𝑅𝑦 → ((∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) ∧ ∃𝑢(𝑢𝑅𝑦𝑢𝑅𝑧)) → ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑧)))
2 brcoss 35556 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑅𝑦 ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)))
32el2v 3499 . . . . . 6 (𝑥𝑅𝑦 ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦))
4 brcoss 35556 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑦𝑢𝑅𝑧)))
54el2v 3499 . . . . . 6 (𝑦𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑦𝑢𝑅𝑧))
63, 5anbi12i 626 . . . . 5 ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) ∧ ∃𝑢(𝑢𝑅𝑦𝑢𝑅𝑧)))
7 brcoss 35556 . . . . . 6 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑧)))
87el2v 3499 . . . . 5 (𝑥𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑧))
91, 6, 83imtr4g 297 . . . 4 (∃*𝑢 𝑢𝑅𝑦 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
109alrimiv 1919 . . 3 (∃*𝑢 𝑢𝑅𝑦 → ∀𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1110alimi 1803 . 2 (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1211alrimiv 1919 1 (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526  wex 1771  ∃*wmo 2613  Vcvv 3492   class class class wbr 5057  ccoss 35334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-coss 35539
This theorem is referenced by: (None)
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