MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dirge Structured version   Visualization version   GIF version

Theorem dirge 17966
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 17963 . . . . . . 7 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
31, 2syl5eq 2786 . . . . . 6 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
43eleq2d 2819 . . . . 5 (𝑅 ∈ DirRel → (𝐴𝑋𝐴 𝑅))
53eleq2d 2819 . . . . 5 (𝑅 ∈ DirRel → (𝐵𝑋𝐵 𝑅))
64, 5anbi12d 634 . . . 4 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) ↔ (𝐴 𝑅𝐵 𝑅)))
7 eqid 2739 . . . . . . . . 9 𝑅 = 𝑅
87isdir 17961 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
98ibi 270 . . . . . . 7 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
109simprrd 774 . . . . . 6 (𝑅 ∈ DirRel → ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))
11 codir 5955 . . . . . 6 (( 𝑅 × 𝑅) ⊆ (𝑅𝑅) ↔ ∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥))
1210, 11sylib 221 . . . . 5 (𝑅 ∈ DirRel → ∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥))
13 breq1 5034 . . . . . . . 8 (𝑦 = 𝐴 → (𝑦𝑅𝑥𝐴𝑅𝑥))
1413anbi1d 633 . . . . . . 7 (𝑦 = 𝐴 → ((𝑦𝑅𝑥𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥𝑧𝑅𝑥)))
1514exbidv 1928 . . . . . 6 (𝑦 = 𝐴 → (∃𝑥(𝑦𝑅𝑥𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥𝑧𝑅𝑥)))
16 breq1 5034 . . . . . . . 8 (𝑧 = 𝐵 → (𝑧𝑅𝑥𝐵𝑅𝑥))
1716anbi2d 632 . . . . . . 7 (𝑧 = 𝐵 → ((𝐴𝑅𝑥𝑧𝑅𝑥) ↔ (𝐴𝑅𝑥𝐵𝑅𝑥)))
1817exbidv 1928 . . . . . 6 (𝑧 = 𝐵 → (∃𝑥(𝐴𝑅𝑥𝑧𝑅𝑥) ↔ ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
1915, 18rspc2v 3537 . . . . 5 ((𝐴 𝑅𝐵 𝑅) → (∀𝑦 𝑅𝑧 𝑅𝑥(𝑦𝑅𝑥𝑧𝑅𝑥) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
2012, 19syl5com 31 . . . 4 (𝑅 ∈ DirRel → ((𝐴 𝑅𝐵 𝑅) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
216, 20sylbid 243 . . 3 (𝑅 ∈ DirRel → ((𝐴𝑋𝐵𝑋) → ∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥)))
22 reldir 17962 . . . . . . . . . 10 (𝑅 ∈ DirRel → Rel 𝑅)
23 relelrn 5789 . . . . . . . . . 10 ((Rel 𝑅𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2422, 23sylan 583 . . . . . . . . 9 ((𝑅 ∈ DirRel ∧ 𝐴𝑅𝑥) → 𝑥 ∈ ran 𝑅)
2524ex 416 . . . . . . . 8 (𝑅 ∈ DirRel → (𝐴𝑅𝑥𝑥 ∈ ran 𝑅))
26 ssun2 4064 . . . . . . . . . . 11 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
27 dmrnssfld 5814 . . . . . . . . . . 11 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
2826, 27sstri 3887 . . . . . . . . . 10 ran 𝑅 𝑅
2928, 3sseqtrrid 3931 . . . . . . . . 9 (𝑅 ∈ DirRel → ran 𝑅𝑋)
3029sseld 3877 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑥 ∈ ran 𝑅𝑥𝑋))
3125, 30syld 47 . . . . . . 7 (𝑅 ∈ DirRel → (𝐴𝑅𝑥𝑥𝑋))
3231adantrd 495 . . . . . 6 (𝑅 ∈ DirRel → ((𝐴𝑅𝑥𝐵𝑅𝑥) → 𝑥𝑋))
3332ancrd 555 . . . . 5 (𝑅 ∈ DirRel → ((𝐴𝑅𝑥𝐵𝑅𝑥) → (𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥))))
3433eximdv 1924 . . . 4 (𝑅 ∈ DirRel → (∃𝑥(𝐴𝑅𝑥𝐵𝑅𝑥) → ∃𝑥(𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥))))
35 df-rex 3060 . . . 4 (∃𝑥𝑋 (𝐴𝑅𝑥𝐵𝑅𝑥) ↔ ∃𝑥(𝑥𝑋 ∧ (𝐴𝑅𝑥𝐵𝑅𝑥)))
3634, 35syl6ibr 255 . . 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 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  wral 3054  wrex 3055  cun 3842  wss 3844   cuni 4797   class class class wbr 5031   I cid 5429   × cxp 5524  ccnv 5525  dom cdm 5526  ran crn 5527  cres 5528  ccom 5530  Rel wrel 5531  DirRelcdir 17957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-dir 17959
This theorem is referenced by:  tailfb  34212
  Copyright terms: Public domain W3C validator