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Theorem codir 6078
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 5673 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2 df-br 5110 . . . . 5 (𝑥(𝑅𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅))
3 brcodir 6077 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
43el2v 3455 . . . . 5 (𝑥(𝑅𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
52, 4bitr3i 277 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
61, 5imbi12i 351 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)) ↔ ((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
762albii 1823 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
8 relxp 5655 . . 3 Rel (𝐴 × 𝐵)
9 ssrel 5742 . . 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 3188 . 2 (∀𝑥𝐴𝑦𝐵𝑧(𝑥𝑅𝑧𝑦𝑅𝑧) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
127, 10, 113bitr4i 303 1 ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wex 1782  wcel 2107  wral 3061  Vcvv 3447  wss 3914  cop 4596   class class class wbr 5109   × cxp 5635  ccnv 5636  ccom 5641  Rel wrel 5642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646
This theorem is referenced by:  dirge  18500  filnetlem3  34905
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