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Theorem tailf 32495
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 5512 . . . . . . 7 (𝐷 “ {𝑥}) ⊆ ran 𝐷
2 ssun2 3810 . . . . . . . 8 ran 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
3 dmrnssfld 5416 . . . . . . . 8 (dom 𝐷 ∪ ran 𝐷) ⊆ 𝐷
42, 3sstri 3645 . . . . . . 7 ran 𝐷 𝐷
51, 4sstri 3645 . . . . . 6 (𝐷 “ {𝑥}) ⊆ 𝐷
6 tailf.1 . . . . . . 7 𝑋 = dom 𝐷
7 dirdm 17281 . . . . . . 7 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
86, 7syl5req 2698 . . . . . 6 (𝐷 ∈ DirRel → 𝐷 = 𝑋)
95, 8syl5sseq 3686 . . . . 5 (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ⊆ 𝑋)
10 dmexg 7139 . . . . . . 7 (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
116, 10syl5eqel 2734 . . . . . 6 (𝐷 ∈ DirRel → 𝑋 ∈ V)
12 elpw2g 4857 . . . . . 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 2996 . . 3 (𝐷 ∈ DirRel → ∀𝑥𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
16 eqid 2651 . . . 4 (𝑥𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥}))
1716fmpt 6421 . . 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 32492 . . 3 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
2019feq1d 6068 . 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 1523  wcel 2030  wral 2941  Vcvv 3231  cun 3605  wss 3607  𝒫 cpw 4191  {csn 4210   cuni 4468  cmpt 4762  dom cdm 5143  ran crn 5144  cima 5146  wf 5922  cfv 5926  DirRelcdir 17275  tailctail 17276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-dir 17277  df-tail 17278
This theorem is referenced by:  tailfb  32497  filnetlem4  32501
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