Theorem codir 6077

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 5661	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2		df-br 5080	. . . . 5 ⊢ (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅))
3		brcodir 6076	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)))
4	3	el2v 3439	. . . . 5 ⊢ (𝑥(◡𝑅 ∘ 𝑅)𝑦 ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))
5	2, 4	bitr3i 278	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅) ↔ ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))
6	1, 5	imbi12i 351	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)))
7	6	2albii 1827	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (◡𝑅 ∘ 𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)))
8		relxp 5643	. . 3 ⊢ Rel (𝐴 × 𝐵)
9		ssrel 5733	. . 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 3176	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧)))
12	7, 10, 11	3bitr4i 304	1 ⊢ ((𝐴 × 𝐵) ⊆ (◡𝑅 ∘ 𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∃𝑧(𝑥𝑅𝑧 ∧ 𝑦𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545  ∃wex 1786   ∈ wcel 2119  ∀wral 3054  Vcvv 3432   ⊆ wss 3890  ⟨cop 4568   class class class wbr 5079   × cxp 5623  ^◡ccnv 5624   ∘ ccom 5629  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634

This theorem is referenced by:  dirge  18567  filnetlem3  36615