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Theorem tailf 36357
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 6090 . . . . . . 7 (𝐷 “ {𝑥}) ⊆ ran 𝐷
2 ssun2 4188 . . . . . . . 8 ran 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
3 dmrnssfld 5986 . . . . . . . 8 (dom 𝐷 ∪ ran 𝐷) ⊆ 𝐷
42, 3sstri 4004 . . . . . . 7 ran 𝐷 𝐷
51, 4sstri 4004 . . . . . 6 (𝐷 “ {𝑥}) ⊆ 𝐷
6 tailf.1 . . . . . . 7 𝑋 = dom 𝐷
7 dirdm 18657 . . . . . . 7 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
86, 7eqtr2id 2787 . . . . . 6 (𝐷 ∈ DirRel → 𝐷 = 𝑋)
95, 8sseqtrid 4047 . . . . 5 (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ⊆ 𝑋)
10 dmexg 7923 . . . . . . 7 (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
116, 10eqeltrid 2842 . . . . . 6 (𝐷 ∈ DirRel → 𝑋 ∈ V)
12 elpw2g 5338 . . . . . 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 3147 . . 3 (𝐷 ∈ DirRel → ∀𝑥𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
16 eqid 2734 . . . 4 (𝑥𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥}))
1716fmpt 7129 . . 3 (∀𝑥𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝑥𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
1815, 17sylib 218 . 2 (𝐷 ∈ DirRel → (𝑥𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
196tailfval 36354 . . 3 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
2019feq1d 6720 . 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 206   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  cun 3960  wss 3962  𝒫 cpw 4604  {csn 4630   cuni 4911  cmpt 5230  dom cdm 5688  ran crn 5689  cima 5691  wf 6558  cfv 6562  DirRelcdir 18651  tailctail 18652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-dir 18653  df-tail 18654
This theorem is referenced by:  tailfb  36359  filnetlem4  36363
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