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Theorem codir 5013
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 4653 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
2 df-br 4001 . . . . 5 (𝑥(𝑅𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅))
3 vex 2740 . . . . . 6 𝑥 ∈ V
4 vex 2740 . . . . . 6 𝑦 ∈ V
5 brcodir 5012 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑅𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
63, 4, 5mp2an 426 . . . . 5 (𝑥(𝑅𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
72, 6bitr3i 186 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅) ↔ ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
81, 7imbi12i 239 . . 3 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)) ↔ ((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
982albii 1471 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝑅𝑅)) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
10 relxp 4732 . . 3 Rel (𝐴 × 𝐵)
11 ssrel 4711 . . 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 2496 . 2 (∀𝑥𝐴𝑦𝐵𝑧(𝑥𝑅𝑧𝑦𝑅𝑧) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → ∃𝑧(𝑥𝑅𝑧𝑦𝑅𝑧)))
149, 12, 133bitr4i 212 1 ((𝐴 × 𝐵) ⊆ (𝑅𝑅) ↔ ∀𝑥𝐴𝑦𝐵𝑧(𝑥𝑅𝑧𝑦𝑅𝑧))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351  wex 1492  wcel 2148  wral 2455  Vcvv 2737  wss 3129  cop 3594   class class class wbr 4000   × cxp 4621  ccnv 4622  ccom 4627  Rel wrel 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632
This theorem is referenced by: (None)
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