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Theorem dirge 17169
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 17166 . . . . . . 7 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
31, 2syl5eq 2667 . . . . . 6 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
43eleq2d 2684 . . . . 5 (𝑅 ∈ DirRel → (𝐴𝑋𝐴 𝑅))
53eleq2d 2684 . . . . 5 (𝑅 ∈ DirRel → (𝐵𝑋𝐵 𝑅))
64, 5anbi12d 746 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) ↔ (𝐴 𝑅𝐵 𝑅)))
7 eqid 2621 . . . . . . . . 9 𝑅 = 𝑅
87isdir 17164 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
98ibi 256 . . . . . . 7 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
109simprrd 796 . . . . . 6 (𝑅 ∈ DirRel → ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))
11 codir 5480 . . . . . 6 (( 𝑅 × 𝑅) ⊆ (𝑅𝑅) ↔ ∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥))
1210, 11sylib 208 . . . . 5 (𝑅 ∈ DirRel → ∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥))
13 breq1 4621 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑅𝑥𝐴𝑅𝑥))
1413anbi1d 740 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦𝑅𝑥𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥𝑧𝑅𝑥)))
1514exbidv 1847 . . . . . 6 (𝑦 = 𝐴 → (∃𝑥(𝑦𝑅𝑥𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥𝑧𝑅𝑥)))
16 breq1 4621 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑅𝑥𝐵𝑅𝑥))
1716anbi2d 739 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴𝑅𝑥𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥𝐵𝑅𝑥)))
1817exbidv 1847 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥(𝐴𝑅𝑥𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
1915, 18rspc2v 3310 . . . . 5 ((𝐴 𝑅𝐵 𝑅) → (∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
2012, 19syl5com 31 . . . 4 (𝑅 ∈ DirRel → ((𝐴 𝑅𝐵 𝑅) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
216, 20sylbid 230 . . 3 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
22 reldir 17165 . . . . . . . . . 10 (𝑅 ∈ DirRel → Rel 𝑅)
23 relelrn 5324 . . . . . . . . . 10 ((Rel 𝑅𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2422, 23sylan 488 . . . . . . . . 9 ((𝑅 ∈ DirRel ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2524ex 450 . . . . . . . 8 (𝑅 ∈ DirRel → (𝐴𝑅𝑥𝑥 ∈ ran 𝑅))
26 ssun2 3760 . . . . . . . . . . 11 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
27 dmrnssfld 5349 . . . . . . . . . . 11 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
2826, 27sstri 3596 . . . . . . . . . 10 ran 𝑅 𝑅
2928, 3syl5sseqr 3638 . . . . . . . . 9 (𝑅 ∈ DirRel → ran 𝑅𝑋)
3029sseld 3586 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑥 ∈ ran 𝑅𝑥𝑋))
3125, 30syld 47 . . . . . . 7 (𝑅 ∈ DirRel → (𝐴𝑅𝑥𝑥𝑋))
3231adantrd 484 . . . . . 6 (𝑅 ∈ DirRel → ((𝐴𝑅𝑥𝐵𝑅𝑥) → 𝑥𝑋))
3332ancrd 576 . . . . 5 (𝑅 ∈ DirRel → ((𝐴𝑅𝑥𝐵𝑅𝑥) → (𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥))))
3433eximdv 1843 . . . 4 (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) → ∃𝑥(𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥))))
35 df-rex 2913 . . . 4 (∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥) ↔ ∃𝑥(𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥)))
3634, 35syl6ibr 242 . . 3 (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥)))
3721, 36syld 47 . 2 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥)))
38373impib 1259 1 ((𝑅 ∈ DirRel ∧ 𝐴𝑋𝐵𝑋) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  cun 3557  wss 3559   cuni 4407   class class class wbr 4618   I cid 4989   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  ccom 5083  Rel wrel 5084  DirRelcdir 17160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-dir 17162
This theorem is referenced by:  tailfb  32049
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