Theorem codir 6113

Description: Two ways of saying a relation is directed. (Contributed by Mario Carneiro, 22-Nov-2013.)

Assertion

Ref	Expression
codir	⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Proof of Theorem codir

Step	Hyp	Ref	Expression
1		opelxp 5686	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		df-br 5103	. . . . 5 ⊢ (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅))
3		brcodir 6112	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)))
4	3	el2v 3457	. . . . 5 ⊢ (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))
5	2, 4	bitr3i 280	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅) ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))
6	1, 5	imbi12i 353	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)))
7	6	2albii 1852	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)))
8		relxp 5668	. . 3 ⊢ Rel (𝐴 × 𝐵)
9		ssrel 5758	. . 3 ⊢ (Rel (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅))))
10	8, 9	ax-mp 5	. 2 ⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅)))
11		r2al 3198	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)))
12	7, 10, 11	3bitr4i 306	1 ⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895  ⟨cop 4834   class class class wbr 5102   × cxp 5648  ^◡ccnv 5649   ∘ ccom 5654  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659

This theorem is used by:  dirge  18782  filnetlem3  37260