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Theorem cotr 5010
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 4635 . . . 4 (𝑅𝑅) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧)}
21relopabi 4752 . . 3 Rel (𝑅𝑅)
3 ssrel 4714 . . 3 (Rel (𝑅𝑅) → ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅)))
42, 3ax-mp 5 . 2 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅))
5 vex 2740 . . . . . . . 8 𝑥 ∈ V
6 vex 2740 . . . . . . . 8 𝑧 ∈ V
75, 6opelco 4799 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) ↔ ∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧))
8 df-br 4004 . . . . . . . 8 (𝑥𝑅𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝑅)
98bicomi 132 . . . . . . 7 (⟨𝑥, 𝑧⟩ ∈ 𝑅𝑥𝑅𝑧)
107, 9imbi12i 239 . . . . . 6 ((⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ (∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
11 19.23v 1883 . . . . . 6 (∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (∃𝑦(𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1210, 11bitr4i 187 . . . . 5 ((⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1312albii 1470 . . . 4 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
14 alcom 1478 . . . 4 (∀𝑧𝑦((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1513, 14bitri 184 . . 3 (∀𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1615albii 1470 . 2 (∀𝑥𝑧(⟨𝑥, 𝑧⟩ ∈ (𝑅𝑅) → ⟨𝑥, 𝑧⟩ ∈ 𝑅) ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
174, 16bitri 184 1 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351  wex 1492  wcel 2148  wss 3129  cop 3595   class class class wbr 4003  ccom 4630  Rel wrel 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-co 4635
This theorem is referenced by:  xpidtr  5019  trin2  5020  dfer2  6535
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