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Theorem dirtr 17157
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 17154 . . . . 5 (𝑅 ∈ DirRel → Rel 𝑅)
2 brrelex 5116 . . . . . . 7 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
32ex 450 . . . . . 6 (Rel 𝑅 → (𝐴𝑅𝐵𝐴 ∈ V))
4 brrelex 5116 . . . . . . 7 ((Rel 𝑅𝐵𝑅𝐶) → 𝐵 ∈ V)
54ex 450 . . . . . 6 (Rel 𝑅 → (𝐵𝑅𝐶𝐵 ∈ V))
63, 5anim12d 585 . . . . 5 (Rel 𝑅 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
71, 6syl 17 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
8 eqid 2621 . . . . . . . . . . . 12 𝑅 = 𝑅
98isdir 17153 . . . . . . . . . . 11 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
109ibi 256 . . . . . . . . . 10 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
1110simprd 479 . . . . . . . . 9 (𝑅 ∈ DirRel → ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))
1211simpld 475 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅𝑅) ⊆ 𝑅)
13 cotr 5467 . . . . . . . 8 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1412, 13sylib 208 . . . . . . 7 (𝑅 ∈ DirRel → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
15 breq12 4618 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑅𝑦𝐴𝑅𝐵))
16153adant3 1079 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑦𝐴𝑅𝐵))
17 breq12 4618 . . . . . . . . . . 11 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
18173adant1 1077 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
1916, 18anbi12d 746 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
20 breq12 4618 . . . . . . . . . 10 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
21203adant2 1078 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
2219, 21imbi12d 334 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2322spc3gv 3284 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶𝑉) → (∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2414, 23syl5 34 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶𝑉) → (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
25243expia 1264 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶𝑉 → (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶))))
2625com4t 93 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐶𝑉𝐴𝑅𝐶))))
277, 26mpdd 43 . . 3 (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐶𝑉𝐴𝑅𝐶)))
2827imp31 448 . 2 (((𝑅 ∈ DirRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) ∧ 𝐶𝑉) → 𝐴𝑅𝐶)
2928an32s 845 1 (((𝑅 ∈ DirRel ∧ 𝐶𝑉) ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  Vcvv 3186  wss 3555   cuni 4402   class class class wbr 4613   I cid 4984   × cxp 5072  ccnv 5073  cres 5076  ccom 5078  Rel wrel 5079  DirRelcdir 17149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-res 5086  df-dir 17151
This theorem is referenced by:  tailfb  32011
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