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

Theorem codir 5699
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 5313 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2 df-br 4810 . . . . 5 (𝑥(𝑅𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅))
3 vex 3353 . . . . . 6 𝑥 ∈ V
4 vex 3353 . . . . . 6 𝑦 ∈ V
5 brcodir 5698 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
63, 4, 5mp2an 683 . . . . 5 (𝑥(𝑅𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
72, 6bitr3i 268 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
81, 7imbi12i 341 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)) ↔ ((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
982albii 1915 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
10 relxp 5295 . . 3 Rel (𝐴 × 𝐵)
11 ssrel 5377 . . 3 (Rel (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅))))
1210, 11ax-mp 5 . 2 ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)))
13 r2al 3086 . 2 (∀𝑥𝐴𝑦𝐵𝑧(𝑥𝑅𝑧𝑦𝑅𝑧) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
149, 12, 133bitr4i 294 1 ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650  wex 1874  wcel 2155  wral 3055  Vcvv 3350  wss 3732  cop 4340   class class class wbr 4809   × cxp 5275  ccnv 5276  ccom 5281  Rel wrel 5282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286
This theorem is referenced by:  dirge  17503  filnetlem3  32818
  Copyright terms: Public domain W3C validator