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Theorem tailf 35198
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 6068 . . . . . . 7 (𝐷 “ {𝑥}) ⊆ ran 𝐷
2 ssun2 4172 . . . . . . . 8 ran 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
3 dmrnssfld 5967 . . . . . . . 8 (dom 𝐷 ∪ ran 𝐷) ⊆ 𝐷
42, 3sstri 3990 . . . . . . 7 ran 𝐷 𝐷
51, 4sstri 3990 . . . . . 6 (𝐷 “ {𝑥}) ⊆ 𝐷
6 tailf.1 . . . . . . 7 𝑋 = dom 𝐷
7 dirdm 18549 . . . . . . 7 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
86, 7eqtr2id 2786 . . . . . 6 (𝐷 ∈ DirRel → 𝐷 = 𝑋)
95, 8sseqtrid 4033 . . . . 5 (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ⊆ 𝑋)
10 dmexg 7889 . . . . . . 7 (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
116, 10eqeltrid 2838 . . . . . 6 (𝐷 ∈ DirRel → 𝑋 ∈ V)
12 elpw2g 5343 . . . . . 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 257 . . . 4 (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
1514ralrimivw 3151 . . 3 (𝐷 ∈ DirRel → ∀𝑥𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
16 eqid 2733 . . . 4 (𝑥𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥}))
1716fmpt 7105 . . 3 (∀𝑥𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝑥𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
1815, 17sylib 217 . 2 (𝐷 ∈ DirRel → (𝑥𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
196tailfval 35195 . . 3 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
2019feq1d 6699 . 2 (𝐷 ∈ DirRel → ((tail‘𝐷):𝑋⟶𝒫 𝑋 ↔ (𝑥𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋))
2118, 20mpbird 257 1 (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cun 3945  wss 3947  𝒫 cpw 4601  {csn 4627   cuni 4907  cmpt 5230  dom cdm 5675  ran crn 5676  cima 5678  wf 6536  cfv 6540  DirRelcdir 18543  tailctail 18544
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-dir 18545  df-tail 18546
This theorem is referenced by:  tailfb  35200  filnetlem4  35204
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