Theorem brcodir 6151

Description: Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.)

Assertion

Ref	Expression
brcodir	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))

Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝑅   𝑧,𝑉   𝑧,𝑊

Proof of Theorem brcodir

Step	Hyp	Ref	Expression
1		brcog 5891	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵)))
2		vex 3492	. . . . . 6 ⊢ 𝑧 ∈ V
3		brcnvg 5904	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝐵 ∈ 𝑊) → (𝑧◡𝑅𝐵 ↔ 𝐵𝑅𝑧))
4	2, 3	mpan 689	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝑧◡𝑅𝐵 ↔ 𝐵𝑅𝑧))
5	4	anbi2d 629	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ((𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ (𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))
6	5	adantl 481	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ (𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))
7	6	exbidv 1920	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑧(𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))
8	1, 7	bitrd 279	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1777   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  ^◡ccnv 5699   ∘ ccom 5704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-cnv 5708  df-co 5709

This theorem is referenced by:  codir  6152