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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 4372 | . . . 4 ⊢ (𝑅 ∘ 𝑅) = {〈𝑥, 𝑧〉 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)} | |
2 | 1 | relopabi 4481 | . . 3 ⊢ Rel (𝑅 ∘ 𝑅) |
3 | ssrel 4446 | . . 3 ⊢ (Rel (𝑅 ∘ 𝑅) → ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅))) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅)) |
5 | vex 2604 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
6 | vex 2604 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
7 | 5, 6 | opelco 4525 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) ↔ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) |
8 | df-br 3786 | . . . . . . . 8 ⊢ (𝑥𝑅𝑧 ↔ 〈𝑥, 𝑧〉 ∈ 𝑅) | |
9 | 8 | bicomi 130 | . . . . . . 7 ⊢ (〈𝑥, 𝑧〉 ∈ 𝑅 ↔ 𝑥𝑅𝑧) |
10 | 7, 9 | imbi12i 237 | . . . . . 6 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
11 | 19.23v 1804 | . . . . . 6 ⊢ (∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ (∃𝑦(𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
12 | 10, 11 | bitr4i 185 | . . . . 5 ⊢ ((〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
13 | 12 | albii 1399 | . . . 4 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑧∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
14 | alcom 1407 | . . . 4 ⊢ (∀𝑧∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | |
15 | 13, 14 | bitri 182 | . . 3 ⊢ (∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
16 | 15 | albii 1399 | . 2 ⊢ (∀𝑥∀𝑧(〈𝑥, 𝑧〉 ∈ (𝑅 ∘ 𝑅) → 〈𝑥, 𝑧〉 ∈ 𝑅) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
17 | 4, 16 | bitri 182 | 1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1282 ∃wex 1421 ∈ wcel 1433 ⊆ wss 2973 〈cop 3401 class class class wbr 3785 ∘ ccom 4367 Rel wrel 4368 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-co 4372 |
This theorem is referenced by: xpidtr 4735 trin2 4736 dfer2 6130 |
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