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Theorem dirtr 17905
 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 17902 . . . . 5 (𝑅 ∈ DirRel → Rel 𝑅)
2 brrelex1 5575 . . . . . . 7 ((Rel 𝑅𝐴𝑅𝐵) → 𝐴 ∈ V)
32ex 417 . . . . . 6 (Rel 𝑅 → (𝐴𝑅𝐵𝐴 ∈ V))
4 brrelex1 5575 . . . . . . 7 ((Rel 𝑅𝐵𝑅𝐶) → 𝐵 ∈ V)
54ex 417 . . . . . 6 (Rel 𝑅 → (𝐵𝑅𝐶𝐵 ∈ V))
63, 5anim12d 612 . . . . 5 (Rel 𝑅 → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
71, 6syl 17 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑅𝐵𝐵𝑅𝐶) → (𝐴 ∈ V ∧ 𝐵 ∈ V)))
8 eqid 2759 . . . . . . . . . . 11 𝑅 = 𝑅
98isdir 17901 . . . . . . . . . 10 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
109ibi 270 . . . . . . . . 9 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
1110simprld 772 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅𝑅) ⊆ 𝑅)
12 cotr 5945 . . . . . . . 8 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1311, 12sylib 221 . . . . . . 7 (𝑅 ∈ DirRel → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
14 breq12 5038 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥𝑅𝑦𝐴𝑅𝐵))
15143adant3 1130 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑦𝐴𝑅𝐵))
16 breq12 5038 . . . . . . . . . . 11 ((𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
17163adant1 1128 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑦𝑅𝑧𝐵𝑅𝐶))
1815, 17anbi12d 634 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝐴𝑅𝐵𝐵𝑅𝐶)))
19 breq12 5038 . . . . . . . . . 10 ((𝑥 = 𝐴𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
20193adant2 1129 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝑥𝑅𝑧𝐴𝑅𝐶))
2118, 20imbi12d 349 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)))
2221spc3gv 3524 . . . . . . 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 1119 . . . . 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 422 . 2 (((𝑅 ∈ DirRel ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) ∧ 𝐶𝑉) → 𝐴𝑅𝐶)
2827an32s 652 1 (((𝑅 ∈ DirRel ∧ 𝐶𝑉) ∧ (𝐴𝑅𝐵𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2112  Vcvv 3410   ⊆ wss 3859  ∪ cuni 4799   class class class wbr 5033   I cid 5430   × cxp 5523  ◡ccnv 5524   ↾ cres 5527   ∘ ccom 5529  Rel wrel 5530  DirRelcdir 17897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-res 5537  df-dir 17899 This theorem is referenced by:  tailfb  34108
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