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Theorem dirtr 17435
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 17432 . . . . 5 (𝑅 ∈ DirRel → Rel 𝑅)
2 brrelex 5311 . . . . . . 7 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
32ex 449 . . . . . 6 (Rel 𝑅 → (𝐴𝑅𝐵𝐴 ∈ V))
4 brrelex 5311 . . . . . . 7 ((Rel 𝑅𝐵𝑅𝐶) → 𝐵 ∈ V)
54ex 449 . . . . . 6 (Rel 𝑅 → (𝐵𝑅𝐶𝐵 ∈ V))
63, 5anim12d 587 . . . . 5 (Rel 𝑅 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
71, 6syl 17 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
8 eqid 2758 . . . . . . . . . . . 12 𝑅 = 𝑅
98isdir 17431 . . . . . . . . . . 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 482 . . . . . . . . 9 (𝑅 ∈ DirRel → ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))
1211simpld 477 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅𝑅) ⊆ 𝑅)
13 cotr 5664 . . . . . . . 8 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1412, 13sylib 208 . . . . . . 7 (𝑅 ∈ DirRel → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
15 breq12 4807 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑅𝑦𝐴𝑅𝐵))
16153adant3 1127 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑦𝐴𝑅𝐵))
17 breq12 4807 . . . . . . . . . . 11 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
18173adant1 1125 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
1916, 18anbi12d 749 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
20 breq12 4807 . . . . . . . . . 10 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
21203adant2 1126 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
2219, 21imbi12d 333 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2322spc3gv 3436 . . . . . . 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 1115 . . . . 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 447 . 2 (((𝑅 ∈ DirRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) ∧ 𝐶𝑉) → 𝐴𝑅𝐶)
2928an32s 881 1 (((𝑅 ∈ DirRel ∧ 𝐶𝑉) ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072  wal 1628   = wceq 1630  wcel 2137  Vcvv 3338  wss 3713   cuni 4586   class class class wbr 4802   I cid 5171   × cxp 5262  ccnv 5263  cres 5266  ccom 5268  Rel wrel 5269  DirRelcdir 17427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-res 5276  df-dir 17429
This theorem is referenced by:  tailfb  32676
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