MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  codir Structured version   Visualization version   GIF version

Theorem codir 6121
Description: Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)
Assertion
Ref Expression
codir ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Proof of Theorem codir
StepHypRef Expression
1 opelxp 5712 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2 df-br 5149 . . . . 5 (𝑥(𝑅𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅))
3 brcodir 6120 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
43el2v 3482 . . . . 5 (𝑥(𝑅𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
52, 4bitr3i 276 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
61, 5imbi12i 350 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)) ↔ ((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
762albii 1822 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
8 relxp 5694 . . 3 Rel (𝐴 × 𝐵)
9 ssrel 5782 . . 3 (Rel (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅))))
108, 9ax-mp 5 . 2 ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)))
11 r2al 3194 . 2 (∀𝑥𝐴𝑦𝐵𝑧(𝑥𝑅𝑧𝑦𝑅𝑧) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
127, 10, 113bitr4i 302 1 ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539  wex 1781  wcel 2106  wral 3061  Vcvv 3474  wss 3948  cop 4634   class class class wbr 5148   × cxp 5674  ccnv 5675  ccom 5680  Rel wrel 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685
This theorem is referenced by:  dirge  18555  filnetlem3  35260
  Copyright terms: Public domain W3C validator