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Mirrors > Home > ILE Home > Th. List > cotr | GIF version |
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.) |
Ref | Expression |
---|---|
cotr | ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-co 4635 | . . . 4 ⊢ (𝑅 ∘ 𝑅) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)} | |
2 | 1 | relopabi 4752 | . . 3 ⊢ Rel (𝑅 ∘ 𝑅) |
3 | ssrel 4714 | . . 3 ⊢ (Rel (𝑅 ∘ 𝑅) → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅))) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅)) |
5 | vex 2740 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
6 | vex 2740 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
7 | 5, 6 | opelco 4799 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) ↔ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) |
8 | df-br 4004 | . . . . . . . 8 ⊢ (𝑥𝑅𝑧 ↔ 〈𝑥, 𝑧〉 ∈ 𝑅) | |
9 | 8 | bicomi 132 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ 𝑅 ↔ 𝑥𝑅𝑧) |
10 | 7, 9 | imbi12i 239 | . . . . . 6 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
11 | 19.23v 1883 | . . . . . 6 ⊢ (∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
12 | 10, 11 | bitr4i 187 | . . . . 5 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
13 | 12 | albii 1470 | . . . 4 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑧∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
14 | alcom 1478 | . . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
15 | 13, 14 | bitri 184 | . . 3 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
16 | 15 | albii 1470 | . 2 ⊢ (∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
17 | 4, 16 | bitri 184 | 1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 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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