Mathbox for Jeff Hankins < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tailf Structured version   Visualization version   GIF version

Theorem tailf 34139
 Description: The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
Hypothesis
Ref Expression
tailf.1 𝑋 = dom 𝐷
Assertion
Ref Expression
tailf (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)

Proof of Theorem tailf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 imassrn 5916 . . . . . . 7 (𝐷 “ {𝑥}) ⊆ ran 𝐷
2 ssun2 4080 . . . . . . . 8 ran 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
3 dmrnssfld 5815 . . . . . . . 8 (dom 𝐷 ∪ ran 𝐷) ⊆ 𝐷
42, 3sstri 3903 . . . . . . 7 ran 𝐷 𝐷
51, 4sstri 3903 . . . . . 6 (𝐷 “ {𝑥}) ⊆ 𝐷
6 tailf.1 . . . . . . 7 𝑋 = dom 𝐷
7 dirdm 17915 . . . . . . 7 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
86, 7syl5req 2806 . . . . . 6 (𝐷 ∈ DirRel → 𝐷 = 𝑋)
95, 8sseqtrid 3946 . . . . 5 (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ⊆ 𝑋)
10 dmexg 7618 . . . . . . 7 (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
116, 10eqeltrid 2856 . . . . . 6 (𝐷 ∈ DirRel → 𝑋 ∈ V)
12 elpw2g 5217 . . . . . 6 (𝑋 ∈ V → ((𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝐷 “ {𝑥}) ⊆ 𝑋))
1311, 12syl 17 . . . . 5 (𝐷 ∈ DirRel → ((𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝐷 “ {𝑥}) ⊆ 𝑋))
149, 13mpbird 260 . . . 4 (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
1514ralrimivw 3114 . . 3 (𝐷 ∈ DirRel → ∀𝑥𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
16 eqid 2758 . . . 4 (𝑥𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥}))
1716fmpt 6870 . . 3 (∀𝑥𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝑥𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
1815, 17sylib 221 . 2 (𝐷 ∈ DirRel → (𝑥𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
196tailfval 34136 . . 3 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
2019feq1d 6487 . 2 (𝐷 ∈ DirRel → ((tail‘𝐷):𝑋⟶𝒫 𝑋 ↔ (𝑥𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋))
2118, 20mpbird 260 1 (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ∪ cun 3858   ⊆ wss 3860  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801   ↦ cmpt 5115  dom cdm 5527  ran crn 5528   “ cima 5530  ⟶wf 6335  ‘cfv 6339  DirRelcdir 17909  tailctail 17910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-dir 17911  df-tail 17912 This theorem is referenced by:  tailfb  34141  filnetlem4  34145
 Copyright terms: Public domain W3C validator