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Theorem tailf 32004
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 5440 . . . . . . 7 (𝐷 “ {𝑥}) ⊆ ran 𝐷
2 ssun2 3760 . . . . . . . 8 ran 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
3 dmrnssfld 5348 . . . . . . . 8 (dom 𝐷 ∪ ran 𝐷) ⊆ 𝐷
42, 3sstri 3597 . . . . . . 7 ran 𝐷 𝐷
51, 4sstri 3597 . . . . . 6 (𝐷 “ {𝑥}) ⊆ 𝐷
6 tailf.1 . . . . . . 7 𝑋 = dom 𝐷
7 dirdm 17150 . . . . . . 7 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
86, 7syl5req 2673 . . . . . 6 (𝐷 ∈ DirRel → 𝐷 = 𝑋)
95, 8syl5sseq 3637 . . . . 5 (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ⊆ 𝑋)
10 dmexg 7045 . . . . . . 7 (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
116, 10syl5eqel 2708 . . . . . 6 (𝐷 ∈ DirRel → 𝑋 ∈ V)
12 elpw2g 4792 . . . . . 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 247 . . . 4 (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
1514ralrimivw 2966 . . 3 (𝐷 ∈ DirRel → ∀𝑥𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
16 eqid 2626 . . . 4 (𝑥𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥}))
1716fmpt 6338 . . 3 (∀𝑥𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝑥𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
1815, 17sylib 208 . 2 (𝐷 ∈ DirRel → (𝑥𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
196tailfval 32001 . . 3 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
2019feq1d 5989 . 2 (𝐷 ∈ DirRel → ((tail‘𝐷):𝑋⟶𝒫 𝑋 ↔ (𝑥𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋))
2118, 20mpbird 247 1 (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1992  wral 2912  Vcvv 3191  cun 3558  wss 3560  𝒫 cpw 4135  {csn 4153   cuni 4407  cmpt 4678  dom cdm 5079  ran crn 5080  cima 5082  wf 5846  cfv 5850  DirRelcdir 17144  tailctail 17145
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  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-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-dir 17146  df-tail 17147
This theorem is referenced by:  tailfb  32006  filnetlem4  32010
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