Theorem tailf 33723

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 5939	. . . . . . 7 ⊢ (𝐷 “ {𝑥}) ⊆ ran 𝐷
2		ssun2 4148	. . . . . . . 8 ⊢ ran 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
3		dmrnssfld 5840	. . . . . . . 8 ⊢ (dom 𝐷 ∪ ran 𝐷) ⊆ ∪ ∪ 𝐷
4	2, 3	sstri 3975	. . . . . . 7 ⊢ ran 𝐷 ⊆ ∪ ∪ 𝐷
5	1, 4	sstri 3975	. . . . . 6 ⊢ (𝐷 “ {𝑥}) ⊆ ∪ ∪ 𝐷
6		tailf.1	. . . . . . 7 ⊢ 𝑋 = dom 𝐷
7		dirdm 17843	. . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪ ∪ 𝐷)
8	6, 7	syl5req 2869	. . . . . 6 ⊢ (𝐷 ∈ DirRel → ∪ ∪ 𝐷 = 𝑋)
9	5, 8	sseqtrid 4018	. . . . 5 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ⊆ 𝑋)
10		dmexg 7612	. . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 ∈ V)
11	6, 10	eqeltrid 2917	. . . . . 6 ⊢ (𝐷 ∈ DirRel → 𝑋 ∈ V)
12		elpw2g 5246	. . . . . 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 259	. . . 4 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
15	14	ralrimivw 3183	. . 3 ⊢ (𝐷 ∈ DirRel → ∀𝑥 ∈ 𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋)
16		eqid 2821	. . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))
17	16	fmpt 6873	. . 3 ⊢ (∀𝑥 ∈ 𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
18	15, 17	sylib 220	. 2 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)
19	6	tailfval 33720	. . 3 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))
20	19	feq1d 6498	. 2 ⊢ (𝐷 ∈ DirRel → ((tail‘𝐷):𝑋⟶𝒫 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋))
21	18, 20	mpbird 259	1 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494   ∪ cun 3933   ⊆ wss 3935  𝒫 cpw 4538  {csn 4566  ∪ cuni 4837   ↦ cmpt 5145  dom cdm 5554  ran crn 5555   “ cima 5557  ⟶wf 6350  ‘cfv 6354  DirRelcdir 17837  tailctail 17838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-dir 17839  df-tail 17840

This theorem is referenced by:  tailfb  33725  filnetlem4  33729