Theorem brcodir 5116

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 4889	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵)))
2		vex 2802	. . . . . 6 ⊢ 𝑧 ∈ V
3		brcnvg 4903	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝐵 ∈ 𝑊) → (𝑧◡𝑅𝐵 ↔ 𝐵𝑅𝑧))
4	2, 3	mpan 424	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝑧◡𝑅𝐵 ↔ 𝐵𝑅𝑧))
5	4	anbi2d 464	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ((𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ (𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))
6	5	adantl 277	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ (𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))
7	6	exbidv 1871	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑧(𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))
8	1, 7	bitrd 188	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∃wex 1538   ∈ wcel 2200  Vcvv 2799   class class class wbr 4083  ^◡ccnv 4718   ∘ ccom 4723

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-cnv 4727  df-co 4728

This theorem is referenced by:  codir  5117