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Theorem dirtr 17460
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 17457 . . . . 5 (𝑅 ∈ DirRel → Rel 𝑅)
2 brrelex 5369 . . . . . . 7 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
32ex 399 . . . . . 6 (Rel 𝑅 → (𝐴𝑅𝐵𝐴 ∈ V))
4 brrelex 5369 . . . . . . 7 ((Rel 𝑅𝐵𝑅𝐶) → 𝐵 ∈ V)
54ex 399 . . . . . 6 (Rel 𝑅 → (𝐵𝑅𝐶𝐵 ∈ V))
63, 5anim12d 598 . . . . 5 (Rel 𝑅 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
71, 6syl 17 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
8 eqid 2817 . . . . . . . . . . . 12 𝑅 = 𝑅
98isdir 17456 . . . . . . . . . . 11 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
109ibi 258 . . . . . . . . . 10 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
1110simprd 485 . . . . . . . . 9 (𝑅 ∈ DirRel → ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))
1211simpld 484 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅𝑅) ⊆ 𝑅)
13 cotr 5731 . . . . . . . 8 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1412, 13sylib 209 . . . . . . 7 (𝑅 ∈ DirRel → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
15 breq12 4860 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑅𝑦𝐴𝑅𝐵))
16153adant3 1155 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑦𝐴𝑅𝐵))
17 breq12 4860 . . . . . . . . . . 11 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
18173adant1 1153 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
1916, 18anbi12d 618 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
20 breq12 4860 . . . . . . . . . 10 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
21203adant2 1154 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
2219, 21imbi12d 335 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2322spc3gv 3502 . . . . . . 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 1143 . . . . 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 406 . 2 (((𝑅 ∈ DirRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) ∧ 𝐶𝑉) → 𝐴𝑅𝐶)
2928an32s 634 1 (((𝑅 ∈ DirRel ∧ 𝐶𝑉) ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100  wal 1635   = wceq 1637  wcel 2157  Vcvv 3402  wss 3780   cuni 4641   class class class wbr 4855   I cid 5231   × cxp 5322  ccnv 5323  cres 5326  ccom 5328  Rel wrel 5329  DirRelcdir 17452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-res 5336  df-dir 17454
This theorem is referenced by:  tailfb  32714
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