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Mirrors > Home > ILE Home > Th. List > codir | GIF version |
Description: Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.) |
Ref | Expression |
---|---|
codir | ⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 4653 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | df-br 4001 | . . . . 5 ⊢ (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (◡𝑅 ∘ 𝑅)) | |
3 | vex 2740 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 2740 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | brcodir 5012 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) | |
6 | 3, 4, 5 | mp2an 426 | . . . . 5 ⊢ (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)) |
7 | 2, 6 | bitr3i 186 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝑅 ∘ 𝑅) ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)) |
8 | 1, 7 | imbi12i 239 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (◡𝑅 ∘ 𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) |
9 | 8 | 2albii 1471 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (◡𝑅 ∘ 𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) |
10 | relxp 4732 | . . 3 ⊢ Rel (𝐴 × 𝐵) | |
11 | ssrel 4711 | . . 3 ⊢ (Rel (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (◡𝑅 ∘ 𝑅)))) | |
12 | 10, 11 | ax-mp 5 | . 2 ⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (◡𝑅 ∘ 𝑅))) |
13 | r2al 2496 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) | |
14 | 9, 12, 13 | 3bitr4i 212 | 1 ⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1351 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 Vcvv 2737 ⊆ wss 3129 〈cop 3594 class class class wbr 4000 × cxp 4621 ◡ccnv 4622 ∘ ccom 4627 Rel wrel 4628 |
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 4118 ax-pow 4171 ax-pr 4206 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 |
This theorem is referenced by: (None) |
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