Theorem dirtr 18525

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 18522	. . . . 5 ⊢ (𝑅 ∈ DirRel → Rel 𝑅)
2		brrelex1 5677	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ V)
3	2	ex 412	. . . . . 6 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐴 ∈ V))
4		brrelex1 5677	. . . . . . 7 ⊢ ((Rel 𝑅 ∧ 𝐵𝑅𝐶) → 𝐵 ∈ V)
5	4	ex 412	. . . . . 6 ⊢ (Rel 𝑅 → (𝐵𝑅𝐶 → 𝐵 ∈ V))
6	3, 5	anim12d 609	. . . . 5 ⊢ (Rel 𝑅 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
7	1, 6	syl 17	. . . 4 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
8		eqid 2736	. . . . . . . . . . 11 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
9	8	isdir 18521	. . . . . . . . . 10 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))))
10	9	ibi 267	. . . . . . . . 9 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))
11	10	simprld 771	. . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
12		cotr 6069	. . . . . . . 8 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
13	11, 12	sylib 218	. . . . . . 7 ⊢ (𝑅 ∈ DirRel → ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
14		breq12 5103	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐵))
15	14	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝐵))
16		breq12 5103	. . . . . . . . . . 11 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦𝑅𝑧 ↔ 𝐵𝑅𝐶))
17	16	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑦𝑅𝑧 ↔ 𝐵𝑅𝐶))
18	15, 17	anbi12d 632	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)))
19		breq12 5103	. . . . . . . . . 10 ⊢ ((𝑥 = 𝐴 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝐶))
20	19	3adant2 1131	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝑥𝑅𝑧 ↔ 𝐴𝑅𝐶))
21	18, 20	imbi12d 344	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
22	21	spc3gv 3558	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ 𝑉) → (∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
23	13, 22	syl5 34	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ 𝑉) → (𝑅 ∈ DirRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
24	23	3expia 1121	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ 𝑉 → (𝑅 ∈ DirRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
25	24	com4t 93	. . . 4 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ∈ 𝑉 → 𝐴𝑅𝐶))))
26	7, 25	mpdd 43	. . 3 ⊢ (𝑅 ∈ DirRel → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → (𝐶 ∈ 𝑉 → 𝐴𝑅𝐶)))
27	26	imp31 417	. 2 ⊢ (((𝑅 ∈ DirRel ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) ∧ 𝐶 ∈ 𝑉) → 𝐴𝑅𝐶)
28	27	an32s 652	1 ⊢ (((𝑅 ∈ DirRel ∧ 𝐶 ∈ 𝑉) ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴𝑅𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901  ∪ cuni 4863   class class class wbr 5098   I cid 5518   × cxp 5622  ^◡ccnv 5623   ↾ cres 5626   ∘ ccom 5628  Rel wrel 5629  DirRelcdir 18517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-res 5636  df-dir 18519

This theorem is referenced by:  tailfb  36571