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Theorem dirtr 17680
 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 ∧ 𝐶𝑉) ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)

Proof of Theorem dirtr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldir 17677 . . . . 5 (𝑅 ∈ DirRel → Rel 𝑅)
2 brrelex1 5496 . . . . . . 7 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
32ex 413 . . . . . 6 (Rel 𝑅 → (𝐴𝑅𝐵𝐴 ∈ V))
4 brrelex1 5496 . . . . . . 7 ((Rel 𝑅𝐵𝑅𝐶) → 𝐵 ∈ V)
54ex 413 . . . . . 6 (Rel 𝑅 → (𝐵𝑅𝐶𝐵 ∈ V))
63, 5anim12d 608 . . . . 5 (Rel 𝑅 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
71, 6syl 17 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
8 eqid 2795 . . . . . . . . . . 11 𝑅 = 𝑅
98isdir 17676 . . . . . . . . . 10 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
109ibi 268 . . . . . . . . 9 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
1110simprld 768 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅𝑅) ⊆ 𝑅)
12 cotr 5853 . . . . . . . 8 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1311, 12sylib 219 . . . . . . 7 (𝑅 ∈ DirRel → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
14 breq12 4971 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑅𝑦𝐴𝑅𝐵))
15143adant3 1125 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑦𝐴𝑅𝐵))
16 breq12 4971 . . . . . . . . . . 11 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
17163adant1 1123 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
1815, 17anbi12d 630 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
19 breq12 4971 . . . . . . . . . 10 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
20193adant2 1124 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
2118, 20imbi12d 346 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2221spc3gv 3547 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶𝑉) → (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2313, 22syl5 34 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶𝑉) → (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
24233expia 1114 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶𝑉 → (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
2524com4t 93 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶𝑉𝐴𝑅𝐶))))
267, 25mpdd 43 . . 3 (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐶𝑉𝐴𝑅𝐶)))
2726imp31 418 . 2 (((𝑅 ∈ DirRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) ∧ 𝐶𝑉) → 𝐴𝑅𝐶)
2827an32s 648 1 (((𝑅 ∈ DirRel ∧ 𝐶𝑉) ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080  ∀wal 1520   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ⊆ wss 3863  ∪ cuni 4749   class class class wbr 4966   I cid 5352   × cxp 5446  ◡ccnv 5447   ↾ cres 5450   ∘ ccom 5452  Rel wrel 5453  DirRelcdir 17672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pr 5226 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-br 4967  df-opab 5029  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-res 5460  df-dir 17674 This theorem is referenced by:  tailfb  33340
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