Theorem tailf 34139

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 5916	. . . . . . 7 ⊢ (𝐷 “ {𝑥}) ⊆ ran 𝐷
2		ssun2 4080	. . . . . . . 8 ⊢ ran 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
3		dmrnssfld 5815	. . . . . . . 8 ⊢ (dom 𝐷 ∪ ran 𝐷) ⊆ ∪ ∪ 𝐷
4	2, 3	sstri 3903	. . . . . . 7 ⊢ ran 𝐷 ⊆ ∪ ∪ 𝐷
5	1, 4	sstri 3903	. . . . . 6 ⊢ (𝐷 “ {𝑥}) ⊆ ∪ ∪ 𝐷
6		tailf.1	. . . . . . 7 ⊢ 𝑋 = dom 𝐷
7		dirdm 17915	. . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪ ∪ 𝐷)
8	6, 7	syl5req 2806	. . . . . 6 ⊢ (𝐷 ∈ DirRel → ∪ ∪ 𝐷 = 𝑋)
9	5, 8	sseqtrid 3946	. . . . 5 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ⊆ 𝑋)
10		dmexg 7618	. . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
11	6, 10	eqeltrid 2856	. . . . . 6 ⊢ (𝐷 ∈ DirRel → 𝑋 ∈ V)
12		elpw2g 5217	. . . . . 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 260	. . . 4 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
15	14	ralrimivw 3114	. . 3 ⊢ (𝐷 ∈ DirRel → ∀𝑥 ∈ 𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
16		eqid 2758	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))
17	16	fmpt 6870	. . 3 ⊢ (∀𝑥 ∈ 𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
18	15, 17	sylib 221	. 2 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
19	6	tailfval 34136	. . 3 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))
20	19	feq1d 6487	. 2 ⊢ (𝐷 ∈ DirRel → ((tail‘𝐷):𝑋⟶𝒫 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋))
21	18, 20	mpbird 260	1 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ∪ cun 3858   ⊆ wss 3860  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801   ↦ cmpt 5115  dom cdm 5527  ran crn 5528   “ cima 5530  ⟶wf 6335  ‘cfv 6339  DirRelcdir 17909  tailctail 17910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-dir 17911  df-tail 17912

This theorem is referenced by:  tailfb  34141  filnetlem4  34145