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Theorem cotr 4736
Description: Two ways of saying a relation is transitive. Definition of transitivity in [Schechter] p. 51. (Contributed by NM, 27-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cotr ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Distinct variable group:   𝑥,𝑦,𝑧,𝑅

Proof of Theorem cotr
StepHypRef Expression
1 df-co 4380 . . . 4 (𝑅𝑅) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧)}
21relopabi 4491 . . 3 Rel (𝑅𝑅)
3 ssrel 4454 . . 3 (Rel (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
42, 3ax-mp 7 . 2 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
5 vex 2605 . . . . . . . 8 𝑥 ∈ V
6 vex 2605 . . . . . . . 8 𝑧 ∈ V
75, 6opelco 4535 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧))
8 df-br 3794 . . . . . . . 8 (𝑥𝑅𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝑅)
98bicomi 130 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ 𝑅𝑥𝑅𝑧)
107, 9imbi12i 237 . . . . . 6 ((⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ (∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
11 19.23v 1805 . . . . . 6 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1210, 11bitr4i 185 . . . . 5 ((⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1312albii 1400 . . . 4 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
14 alcom 1408 . . . 4 (∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1513, 14bitri 182 . . 3 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1615albii 1400 . 2 (∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
174, 16bitri 182 1 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wex 1422  wcel 1434  wss 2974  cop 3409   class class class wbr 3793  ccom 4375  Rel wrel 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-co 4380
This theorem is referenced by:  xpidtr  4745  trin2  4746  dfer2  6173
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