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Theorem dirge 18648
Description: For any two elements of a directed set, there exists a third element greater than or equal to both. Note that this does not say that the two elements have a least upper bound. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Hypothesis
Ref Expression
dirge.1 𝑋 = dom 𝑅
Assertion
Ref Expression
dirge ((𝑅 ∈ DirRel ∧ 𝐴𝑋𝐵𝑋) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑋

Proof of Theorem dirge
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dirge.1 . . . . . . 7 𝑋 = dom 𝑅
2 dirdm 18645 . . . . . . 7 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
31, 2eqtrid 2789 . . . . . 6 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
43eleq2d 2827 . . . . 5 (𝑅 ∈ DirRel → (𝐴𝑋𝐴 𝑅))
53eleq2d 2827 . . . . 5 (𝑅 ∈ DirRel → (𝐵𝑋𝐵 𝑅))
64, 5anbi12d 632 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) ↔ (𝐴 𝑅𝐵 𝑅)))
7 eqid 2737 . . . . . . . . 9 𝑅 = 𝑅
87isdir 18643 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
98ibi 267 . . . . . . 7 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
109simprrd 774 . . . . . 6 (𝑅 ∈ DirRel → ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))
11 codir 6140 . . . . . 6 (( 𝑅 × 𝑅) ⊆ (𝑅𝑅) ↔ ∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥))
1210, 11sylib 218 . . . . 5 (𝑅 ∈ DirRel → ∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥))
13 breq1 5146 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑅𝑥𝐴𝑅𝑥))
1413anbi1d 631 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦𝑅𝑥𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥𝑧𝑅𝑥)))
1514exbidv 1921 . . . . . 6 (𝑦 = 𝐴 → (∃𝑥(𝑦𝑅𝑥𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥𝑧𝑅𝑥)))
16 breq1 5146 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑅𝑥𝐵𝑅𝑥))
1716anbi2d 630 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴𝑅𝑥𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥𝐵𝑅𝑥)))
1817exbidv 1921 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥(𝐴𝑅𝑥𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
1915, 18rspc2v 3633 . . . . 5 ((𝐴 𝑅𝐵 𝑅) → (∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
2012, 19syl5com 31 . . . 4 (𝑅 ∈ DirRel → ((𝐴 𝑅𝐵 𝑅) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
216, 20sylbid 240 . . 3 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
22 reldir 18644 . . . . . . . . . 10 (𝑅 ∈ DirRel → Rel 𝑅)
23 relelrn 5956 . . . . . . . . . 10 ((Rel 𝑅𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2422, 23sylan 580 . . . . . . . . 9 ((𝑅 ∈ DirRel ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2524ex 412 . . . . . . . 8 (𝑅 ∈ DirRel → (𝐴𝑅𝑥𝑥 ∈ ran 𝑅))
26 ssun2 4179 . . . . . . . . . . 11 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
27 dmrnssfld 5984 . . . . . . . . . . 11 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
2826, 27sstri 3993 . . . . . . . . . 10 ran 𝑅 𝑅
2928, 3sseqtrrid 4027 . . . . . . . . 9 (𝑅 ∈ DirRel → ran 𝑅𝑋)
3029sseld 3982 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑥 ∈ ran 𝑅𝑥𝑋))
3125, 30syld 47 . . . . . . 7 (𝑅 ∈ DirRel → (𝐴𝑅𝑥𝑥𝑋))
3231adantrd 491 . . . . . 6 (𝑅 ∈ DirRel → ((𝐴𝑅𝑥𝐵𝑅𝑥) → 𝑥𝑋))
3332ancrd 551 . . . . 5 (𝑅 ∈ DirRel → ((𝐴𝑅𝑥𝐵𝑅𝑥) → (𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥))))
3433eximdv 1917 . . . 4 (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) → ∃𝑥(𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥))))
35 df-rex 3071 . . . 4 (∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥) ↔ ∃𝑥(𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥)))
3634, 35imbitrrdi 252 . . 3 (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥)))
3721, 36syld 47 . 2 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥)))
38373impib 1117 1 ((𝑅 ∈ DirRel ∧ 𝐴𝑋𝐵𝑋) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  cun 3949  wss 3951   cuni 4907   class class class wbr 5143   I cid 5577   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  ccom 5689  Rel wrel 5690  DirRelcdir 18639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-dir 18641
This theorem is referenced by:  tailfb  36378
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