Theorem dirtr 18653

Description: A direction is transitive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

Assertion

Ref	Expression
dirtr	⊢ (((𝑅 ∈ DirRel ∧ 𝐶 ∈ 𝑉) ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴𝑅𝐶)

Proof of Theorem dirtr

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldir 18650	. . . . 5 ⊢ (𝑅 ∈ DirRel → Rel 𝑅)
2		brrelex1 5714	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
3	2	ex 417	. . . . . 6 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐴 ∈ V))
4		brrelex1 5714	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐵 ∈ V)
5	4	ex 417	. . . . . 6 ⊢ (Rel 𝑅 → (𝐵𝑅𝐶 → 𝐵 ∈ V))
6	3, 5	anim12d 620	. . . . 5 ⊢ (Rel 𝑅 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
7	1, 6	syl 18	. . . 4 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
8		eqid 2763	. . . . . . . . . . 11 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
9	8	isdir 18649	. . . . . . . . . 10 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))))
10	9	ibi 270	. . . . . . . . 9 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))
11	10	simprld 783	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
12		cotr 6112	. . . . . . . 8 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
13	11, 12	sylib 221	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
14		breq12 5114	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐵))
15	14	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐵))
16		breq12 5114	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦𝑅𝑧 ↔ 𝐵𝑅𝐶))
17	16	3adant1 1148	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦𝑅𝑧 ↔ 𝐵𝑅𝐶))
18	15, 17	anbi12d 643	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)))
19		breq12 5114	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝐶))
20	19	3adant2 1149	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝐶))
21	18, 20	imbi12d 347	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
22	21	spc3gv 3563	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ 𝑉) → (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
23	13, 22	syl5 35	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ 𝑉) → (𝑅 ∈ DirRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
24	23	3expia 1139	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ 𝑉 → (𝑅 ∈ DirRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
25	24	com4t 94	. . . 4 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ 𝑉 → 𝐴𝑅𝐶))))
26	7, 25	mpdd 44	. . 3 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → (𝐶 ∈ 𝑉 → 𝐴𝑅𝐶)))
27	26	imp31 422	. 2 ⊢ (((𝑅 ∈ DirRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) ∧ 𝐶 ∈ 𝑉) → 𝐴𝑅𝐶)
28	27	an32s 664	1 ⊢ (((𝑅 ∈ DirRel ∧ 𝐶 ∈ 𝑉) ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  ∪ cuni 4872   class class class wbr 5109   I cid 5555   × cxp 5659  ^◡ccnv 5660   ↾ cres 5663   ∘ ccom 5665  Rel wrel 5666  DirRelcdir 18645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-res 5673  df-dir 18647

This theorem is referenced by:  tailfb  36888