Theorem trcoss 39031

Description: Sufficient condition for the transitivity of cosets by 𝑅. (Contributed by Peter Mazsa, 26-Dec-2018.)

Assertion

Ref	Expression
trcoss	⊢ (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))

Distinct variable groups:   𝑢,𝑅,𝑥   𝑧,𝑅,𝑢   𝑦,𝑢,𝑥   𝑦,𝑧

Allowed substitution hint:   𝑅(𝑦)

Proof of Theorem trcoss

Step	Hyp	Ref	Expression
1		moantr 38831	. . . . 5 ⊢ (∃*𝑢 𝑢𝑅𝑦 → ((∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ∧ ∃𝑢(𝑢𝑅𝑦 ∧ 𝑢𝑅𝑧)) → ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑧)))
2		brcoss 38980	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ 𝑅𝑦 ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)))
3	2	el2v 3460	. . . . . 6 ⊢ (𝑥 ≀ 𝑅𝑦 ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
4		brcoss 38980	. . . . . . 7 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (𝑦 ≀ 𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑦 ∧ 𝑢𝑅𝑧)))
5	4	el2v 3460	. . . . . 6 ⊢ (𝑦 ≀ 𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑦 ∧ 𝑢𝑅𝑧))
6	3, 5	anbi12i 637	. . . . 5 ⊢ ((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) ↔ (∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦) ∧ ∃𝑢(𝑢𝑅𝑦 ∧ 𝑢𝑅𝑧)))
7		brcoss 38980	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑧 ∈ V) → (𝑥 ≀ 𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑧)))
8	7	el2v 3460	. . . . 5 ⊢ (𝑥 ≀ 𝑅𝑧 ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑧))
9	1, 6, 8	3imtr4g 298	. . . 4 ⊢ (∃*𝑢 𝑢𝑅𝑦 → ((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
10	9	alrimiv 1946	. . 3 ⊢ (∃*𝑢 𝑢𝑅𝑦 → ∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
11	10	alimi 1830	. 2 ⊢ (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))
12	11	alrimiv 1946	1 ⊢ (∀𝑦∃*𝑢 𝑢𝑅𝑦 → ∀𝑥∀𝑦∀𝑧((𝑥 ≀ 𝑅𝑦 ∧ 𝑦 ≀ 𝑅𝑧) → 𝑥 ≀ 𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557  ∃wex 1798  ∃*wmo 2563  Vcvv 3453   class class class wbr 5097   ≀ ccoss 38642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-coss 38960

This theorem is referenced by:  disjim  39343