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Theorem trcoss 35875
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 35769 . . . . 5 (∃*𝑢 𝑢𝑅𝑦 → ((∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) ∧ ∃𝑢(𝑢𝑅𝑦𝑢𝑅𝑧)) → ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑧)))
2 brcoss 35829 . . . . . . 7 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑅𝑦 ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)))
32el2v 3451 . . . . . 6 (𝑥𝑅𝑦 ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦))
4 brcoss 35829 . . . . . . 7 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑦𝑢𝑅𝑧)))
54el2v 3451 . . . . . 6 (𝑦𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑦𝑢𝑅𝑧))
63, 5anbi12i 629 . . . . 5 ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) ∧ ∃𝑢(𝑢𝑅𝑦𝑢𝑅𝑧)))
7 brcoss 35829 . . . . . 6 ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑧)))
87el2v 3451 . . . . 5 (𝑥𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑧))
91, 6, 83imtr4g 299 . . . 4 (∃*𝑢 𝑢𝑅𝑦 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
109alrimiv 1928 . . 3 (∃*𝑢 𝑢𝑅𝑦 → ∀𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1110alimi 1813 . 2 (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1211alrimiv 1928 1 (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781  ∃*wmo 2599  Vcvv 3444   class class class wbr 5033  ccoss 35606
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-coss 35812
This theorem is referenced by: (None)
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