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Theorem dirge 18321
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 18318 . . . . . . 7 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
31, 2eqtrid 2790 . . . . . 6 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
43eleq2d 2824 . . . . 5 (𝑅 ∈ DirRel → (𝐴𝑋𝐴 𝑅))
53eleq2d 2824 . . . . 5 (𝑅 ∈ DirRel → (𝐵𝑋𝐵 𝑅))
64, 5anbi12d 631 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) ↔ (𝐴 𝑅𝐵 𝑅)))
7 eqid 2738 . . . . . . . . 9 𝑅 = 𝑅
87isdir 18316 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
98ibi 266 . . . . . . 7 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
109simprrd 771 . . . . . 6 (𝑅 ∈ DirRel → ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))
11 codir 6025 . . . . . 6 (( 𝑅 × 𝑅) ⊆ (𝑅𝑅) ↔ ∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥))
1210, 11sylib 217 . . . . 5 (𝑅 ∈ DirRel → ∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥))
13 breq1 5077 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑅𝑥𝐴𝑅𝑥))
1413anbi1d 630 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦𝑅𝑥𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥𝑧𝑅𝑥)))
1514exbidv 1924 . . . . . 6 (𝑦 = 𝐴 → (∃𝑥(𝑦𝑅𝑥𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥𝑧𝑅𝑥)))
16 breq1 5077 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑅𝑥𝐵𝑅𝑥))
1716anbi2d 629 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴𝑅𝑥𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥𝐵𝑅𝑥)))
1817exbidv 1924 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥(𝐴𝑅𝑥𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
1915, 18rspc2v 3570 . . . . 5 ((𝐴 𝑅𝐵 𝑅) → (∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
2012, 19syl5com 31 . . . 4 (𝑅 ∈ DirRel → ((𝐴 𝑅𝐵 𝑅) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
216, 20sylbid 239 . . 3 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
22 reldir 18317 . . . . . . . . . 10 (𝑅 ∈ DirRel → Rel 𝑅)
23 relelrn 5854 . . . . . . . . . 10 ((Rel 𝑅𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2422, 23sylan 580 . . . . . . . . 9 ((𝑅 ∈ DirRel ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2524ex 413 . . . . . . . 8 (𝑅 ∈ DirRel → (𝐴𝑅𝑥𝑥 ∈ ran 𝑅))
26 ssun2 4107 . . . . . . . . . . 11 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
27 dmrnssfld 5879 . . . . . . . . . . 11 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
2826, 27sstri 3930 . . . . . . . . . 10 ran 𝑅 𝑅
2928, 3sseqtrrid 3974 . . . . . . . . 9 (𝑅 ∈ DirRel → ran 𝑅𝑋)
3029sseld 3920 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑥 ∈ ran 𝑅𝑥𝑋))
3125, 30syld 47 . . . . . . 7 (𝑅 ∈ DirRel → (𝐴𝑅𝑥𝑥𝑋))
3231adantrd 492 . . . . . 6 (𝑅 ∈ DirRel → ((𝐴𝑅𝑥𝐵𝑅𝑥) → 𝑥𝑋))
3332ancrd 552 . . . . 5 (𝑅 ∈ DirRel → ((𝐴𝑅𝑥𝐵𝑅𝑥) → (𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥))))
3433eximdv 1920 . . . 4 (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) → ∃𝑥(𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥))))
35 df-rex 3070 . . . 4 (∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥) ↔ ∃𝑥(𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥)))
3634, 35syl6ibr 251 . . 3 (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥)))
3721, 36syld 47 . 2 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥)))
38373impib 1115 1 ((𝑅 ∈ DirRel ∧ 𝐴𝑋𝐵𝑋) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  wral 3064  wrex 3065  cun 3885  wss 3887   cuni 4839   class class class wbr 5074   I cid 5488   × cxp 5587  ccnv 5588  dom cdm 5589  ran crn 5590  cres 5591  ccom 5593  Rel wrel 5594  DirRelcdir 18312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-dir 18314
This theorem is referenced by:  tailfb  34566
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