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Theorem dirge 17839
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 17836 . . . . . . 7 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
31, 2syl5eq 2866 . . . . . 6 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
43eleq2d 2896 . . . . 5 (𝑅 ∈ DirRel → (𝐴𝑋𝐴 𝑅))
53eleq2d 2896 . . . . 5 (𝑅 ∈ DirRel → (𝐵𝑋𝐵 𝑅))
64, 5anbi12d 632 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) ↔ (𝐴 𝑅𝐵 𝑅)))
7 eqid 2819 . . . . . . . . 9 𝑅 = 𝑅
87isdir 17834 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
98ibi 269 . . . . . . 7 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
109simprrd 772 . . . . . 6 (𝑅 ∈ DirRel → ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))
11 codir 5973 . . . . . 6 (( 𝑅 × 𝑅) ⊆ (𝑅𝑅) ↔ ∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥))
1210, 11sylib 220 . . . . 5 (𝑅 ∈ DirRel → ∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥))
13 breq1 5060 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑅𝑥𝐴𝑅𝑥))
1413anbi1d 631 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦𝑅𝑥𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥𝑧𝑅𝑥)))
1514exbidv 1915 . . . . . 6 (𝑦 = 𝐴 → (∃𝑥(𝑦𝑅𝑥𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥𝑧𝑅𝑥)))
16 breq1 5060 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑅𝑥𝐵𝑅𝑥))
1716anbi2d 630 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴𝑅𝑥𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥𝐵𝑅𝑥)))
1817exbidv 1915 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥(𝐴𝑅𝑥𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
1915, 18rspc2v 3631 . . . . 5 ((𝐴 𝑅𝐵 𝑅) → (∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
2012, 19syl5com 31 . . . 4 (𝑅 ∈ DirRel → ((𝐴 𝑅𝐵 𝑅) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
216, 20sylbid 242 . . 3 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
22 reldir 17835 . . . . . . . . . 10 (𝑅 ∈ DirRel → Rel 𝑅)
23 relelrn 5808 . . . . . . . . . 10 ((Rel 𝑅𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2422, 23sylan 582 . . . . . . . . 9 ((𝑅 ∈ DirRel ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2524ex 415 . . . . . . . 8 (𝑅 ∈ DirRel → (𝐴𝑅𝑥𝑥 ∈ ran 𝑅))
26 ssun2 4147 . . . . . . . . . . 11 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
27 dmrnssfld 5834 . . . . . . . . . . 11 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
2826, 27sstri 3974 . . . . . . . . . 10 ran 𝑅 𝑅
2928, 3sseqtrrid 4018 . . . . . . . . 9 (𝑅 ∈ DirRel → ran 𝑅𝑋)
3029sseld 3964 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑥 ∈ ran 𝑅𝑥𝑋))
3125, 30syld 47 . . . . . . 7 (𝑅 ∈ DirRel → (𝐴𝑅𝑥𝑥𝑋))
3231adantrd 494 . . . . . 6 (𝑅 ∈ DirRel → ((𝐴𝑅𝑥𝐵𝑅𝑥) → 𝑥𝑋))
3332ancrd 554 . . . . 5 (𝑅 ∈ DirRel → ((𝐴𝑅𝑥𝐵𝑅𝑥) → (𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥))))
3433eximdv 1911 . . . 4 (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) → ∃𝑥(𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥))))
35 df-rex 3142 . . . 4 (∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥) ↔ ∃𝑥(𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥)))
3634, 35syl6ibr 254 . . 3 (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥)))
3721, 36syld 47 . 2 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥)))
38373impib 1110 1 ((𝑅 ∈ DirRel ∧ 𝐴𝑋𝐵𝑋) → ∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wral 3136  wrex 3137  cun 3932  wss 3934   cuni 4830   class class class wbr 5057   I cid 5452   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cres 5550  ccom 5552  Rel wrel 5553  DirRelcdir 17830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-dir 17832
This theorem is referenced by:  tailfb  33713
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