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Theorem tailf 35565
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 β†’ (tailβ€˜π·):π‘‹βŸΆπ’« 𝑋)

Proof of Theorem tailf
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 imassrn 6071 . . . . . . 7 (𝐷 β€œ {π‘₯}) βŠ† ran 𝐷
2 ssun2 4174 . . . . . . . 8 ran 𝐷 βŠ† (dom 𝐷 βˆͺ ran 𝐷)
3 dmrnssfld 5970 . . . . . . . 8 (dom 𝐷 βˆͺ ran 𝐷) βŠ† βˆͺ βˆͺ 𝐷
42, 3sstri 3992 . . . . . . 7 ran 𝐷 βŠ† βˆͺ βˆͺ 𝐷
51, 4sstri 3992 . . . . . 6 (𝐷 β€œ {π‘₯}) βŠ† βˆͺ βˆͺ 𝐷
6 tailf.1 . . . . . . 7 𝑋 = dom 𝐷
7 dirdm 18559 . . . . . . 7 (𝐷 ∈ DirRel β†’ dom 𝐷 = βˆͺ βˆͺ 𝐷)
86, 7eqtr2id 2783 . . . . . 6 (𝐷 ∈ DirRel β†’ βˆͺ βˆͺ 𝐷 = 𝑋)
95, 8sseqtrid 4035 . . . . 5 (𝐷 ∈ DirRel β†’ (𝐷 β€œ {π‘₯}) βŠ† 𝑋)
10 dmexg 7898 . . . . . . 7 (𝐷 ∈ DirRel β†’ dom 𝐷 ∈ V)
116, 10eqeltrid 2835 . . . . . 6 (𝐷 ∈ DirRel β†’ 𝑋 ∈ V)
12 elpw2g 5345 . . . . . 6 (𝑋 ∈ V β†’ ((𝐷 β€œ {π‘₯}) ∈ 𝒫 𝑋 ↔ (𝐷 β€œ {π‘₯}) βŠ† 𝑋))
1311, 12syl 17 . . . . 5 (𝐷 ∈ DirRel β†’ ((𝐷 β€œ {π‘₯}) ∈ 𝒫 𝑋 ↔ (𝐷 β€œ {π‘₯}) βŠ† 𝑋))
149, 13mpbird 256 . . . 4 (𝐷 ∈ DirRel β†’ (𝐷 β€œ {π‘₯}) ∈ 𝒫 𝑋)
1514ralrimivw 3148 . . 3 (𝐷 ∈ DirRel β†’ βˆ€π‘₯ ∈ 𝑋 (𝐷 β€œ {π‘₯}) ∈ 𝒫 𝑋)
16 eqid 2730 . . . 4 (π‘₯ ∈ 𝑋 ↦ (𝐷 β€œ {π‘₯})) = (π‘₯ ∈ 𝑋 ↦ (𝐷 β€œ {π‘₯}))
1716fmpt 7112 . . 3 (βˆ€π‘₯ ∈ 𝑋 (𝐷 β€œ {π‘₯}) ∈ 𝒫 𝑋 ↔ (π‘₯ ∈ 𝑋 ↦ (𝐷 β€œ {π‘₯})):π‘‹βŸΆπ’« 𝑋)
1815, 17sylib 217 . 2 (𝐷 ∈ DirRel β†’ (π‘₯ ∈ 𝑋 ↦ (𝐷 β€œ {π‘₯})):π‘‹βŸΆπ’« 𝑋)
196tailfval 35562 . . 3 (𝐷 ∈ DirRel β†’ (tailβ€˜π·) = (π‘₯ ∈ 𝑋 ↦ (𝐷 β€œ {π‘₯})))
2019feq1d 6703 . 2 (𝐷 ∈ DirRel β†’ ((tailβ€˜π·):π‘‹βŸΆπ’« 𝑋 ↔ (π‘₯ ∈ 𝑋 ↦ (𝐷 β€œ {π‘₯})):π‘‹βŸΆπ’« 𝑋))
2118, 20mpbird 256 1 (𝐷 ∈ DirRel β†’ (tailβ€˜π·):π‘‹βŸΆπ’« 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  Vcvv 3472   βˆͺ cun 3947   βŠ† wss 3949  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909   ↦ cmpt 5232  dom cdm 5677  ran crn 5678   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  DirRelcdir 18553  tailctail 18554
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-dir 18555  df-tail 18556
This theorem is referenced by:  tailfb  35567  filnetlem4  35571
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