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Mirrors > Home > MPE Home > Th. List > codir | Structured version Visualization version 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 5702 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | df-br 5139 | . . . . 5 ⊢ (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅)) | |
3 | brcodir 6110 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) | |
4 | 3 | el2v 3474 | . . . . 5 ⊢ (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)) |
5 | 2, 4 | bitr3i 277 | . . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅) ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)) |
6 | 1, 5 | imbi12i 350 | . . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) |
7 | 6 | 2albii 1814 | . 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) |
8 | relxp 5684 | . . 3 ⊢ Rel (𝐴 × 𝐵) | |
9 | ssrel 5772 | . . 3 ⊢ (Rel (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅)))) | |
10 | 8, 9 | ax-mp 5 | . 2 ⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅))) |
11 | r2al 3186 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) | |
12 | 7, 10, 11 | 3bitr4i 303 | 1 ⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 ∃wex 1773 ∈ wcel 2098 ∀wral 3053 Vcvv 3466 ⊆ wss 3940 ⟨cop 4626 class class class wbr 5138 × cxp 5664 ◡ccnv 5665 ∘ ccom 5670 Rel wrel 5671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-br 5139 df-opab 5201 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 |
This theorem is referenced by: dirge 18558 filnetlem3 35755 |
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