Theorem brcodir 6120

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 5866	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵)))
2		vex 3478	. . . . . 6 ⊢ 𝑧 ∈ V
3		brcnvg 5879	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝐵 ∈ 𝑊) → (𝑧◡𝑅𝐵 ↔ 𝐵𝑅𝑧))
4	2, 3	mpan 688	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝑧◡𝑅𝐵 ↔ 𝐵𝑅𝑧))
5	4	anbi2d 629	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ((𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ (𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))
6	5	adantl 482	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ (𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))
7	6	exbidv 1924	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑧(𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))
8	1, 7	bitrd 278	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∃wex 1781   ∈ wcel 2106  Vcvv 3474   class class class wbr 5148  ^◡ccnv 5675   ∘ ccom 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-cnv 5684  df-co 5685

This theorem is referenced by:  codir  6121