Theorem tailf 36376

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.

Step	Hyp	Ref	Expression
1		imassrn 6089	. . . . . . 7 ⊢ (𝐷 “ {𝑥}) ⊆ ran 𝐷
2		ssun2 4179	. . . . . . . 8 ⊢ ran 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
3		dmrnssfld 5984	. . . . . . . 8 ⊢ (dom 𝐷 ∪ ran 𝐷) ⊆ ∪ ∪ 𝐷
4	2, 3	sstri 3993	. . . . . . 7 ⊢ ran 𝐷 ⊆ ∪ ∪ 𝐷
5	1, 4	sstri 3993	. . . . . 6 ⊢ (𝐷 “ {𝑥}) ⊆ ∪ ∪ 𝐷
6		tailf.1	. . . . . . 7 ⊢ 𝑋 = dom 𝐷
7		dirdm 18645	. . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪ ∪ 𝐷)
8	6, 7	eqtr2id 2790	. . . . . 6 ⊢ (𝐷 ∈ DirRel → ∪ ∪ 𝐷 = 𝑋)
9	5, 8	sseqtrid 4026	. . . . 5 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ⊆ 𝑋)
10		dmexg 7923	. . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
11	6, 10	eqeltrid 2845	. . . . . 6 ⊢ (𝐷 ∈ DirRel → 𝑋 ∈ V)
12		elpw2g 5333	. . . . . 6 ⊢ (𝑋 ∈ V → ((𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝐷 “ {𝑥}) ⊆ 𝑋))
13	11, 12	syl 17	. . . . 5 ⊢ (𝐷 ∈ DirRel → ((𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝐷 “ {𝑥}) ⊆ 𝑋))
14	9, 13	mpbird 257	. . . 4 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
15	14	ralrimivw 3150	. . 3 ⊢ (𝐷 ∈ DirRel → ∀𝑥 ∈ 𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
16		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))
17	16	fmpt 7130	. . 3 ⊢ (∀𝑥 ∈ 𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
18	15, 17	sylib 218	. 2 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
19	6	tailfval 36373	. . 3 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))
20	19	feq1d 6720	. 2 ⊢ (𝐷 ∈ DirRel → ((tail‘𝐷):𝑋⟶𝒫 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋))
21	18, 20	mpbird 257	1 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480   ∪ cun 3949   ⊆ wss 3951  𝒫 cpw 4600  {csn 4626  ∪ cuni 4907   ↦ cmpt 5225  dom cdm 5685  ran crn 5686   “ cima 5688  ⟶wf 6557  ‘cfv 6561  DirRelcdir 18639  tailctail 18640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-dir 18641  df-tail 18642

This theorem is referenced by:  tailfb  36378  filnetlem4  36382