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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 4607 | . . . 4 ⊢ (𝑅 ∘ 𝑅) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)} | |
2 | 1 | relopabi 4724 | . . 3 ⊢ Rel (𝑅 ∘ 𝑅) |
3 | ssrel 4686 | . . 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 2724 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
6 | vex 2724 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
7 | 5, 6 | opelco 4770 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) ↔ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) |
8 | df-br 3977 | . . . . . . . 8 ⊢ (𝑥𝑅𝑧 ↔ 〈𝑥, 𝑧〉 ∈ 𝑅) | |
9 | 8 | bicomi 131 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ 𝑅 ↔ 𝑥𝑅𝑧) |
10 | 7, 9 | imbi12i 238 | . . . . . 6 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
11 | 19.23v 1870 | . . . . . 6 ⊢ (∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
12 | 10, 11 | bitr4i 186 | . . . . 5 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
13 | 12 | albii 1457 | . . . 4 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑧∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
14 | alcom 1465 | . . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
15 | 13, 14 | bitri 183 | . . 3 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
16 | 15 | albii 1457 | . 2 ⊢ (∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
17 | 4, 16 | bitri 183 | 1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1340 ∃wex 1479 ∈ wcel 2135 ⊆ wss 3111 〈cop 3573 class class class wbr 3976 ∘ ccom 4602 Rel wrel 4603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-co 4607 |
This theorem is referenced by: xpidtr 4988 trin2 4989 dfer2 6493 |
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