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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 5146 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | df-br 4654 | . . . . 5 ⊢ (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (◡𝑅 ∘ 𝑅)) | |
3 | vex 3203 | . . . . . 6 ⊢ 𝑥 ∈ V | |
4 | vex 3203 | . . . . . 6 ⊢ 𝑦 ∈ V | |
5 | brcodir 5515 | . . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) | |
6 | 3, 4, 5 | mp2an 708 | . . . . 5 ⊢ (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)) |
7 | 2, 6 | bitr3i 266 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (◡𝑅 ∘ 𝑅) ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)) |
8 | 1, 7 | imbi12i 340 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (◡𝑅 ∘ 𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) |
9 | 8 | 2albii 1748 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (◡𝑅 ∘ 𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) |
10 | relxp 5227 | . . 3 ⊢ Rel (𝐴 × 𝐵) | |
11 | ssrel 5207 | . . 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 2939 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))) | |
14 | 9, 12, 13 | 3bitr4i 292 | 1 ⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 〈cop 4183 class class class wbr 4653 × cxp 5112 ◡ccnv 5113 ∘ ccom 5118 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 |
This theorem is referenced by: dirge 17237 filnetlem3 32375 |
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