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Theorem tailfval 36374
Description: The tail function for a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
Hypothesis
Ref Expression
tailfval.1 𝑋 = dom 𝐷
Assertion
Ref Expression
tailfval (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
Distinct variable groups:   𝑥,𝐷   𝑥,𝑋

Proof of Theorem tailfval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 uniexg 7761 . . . 4 (𝐷 ∈ DirRel → 𝐷 ∈ V)
2 uniexg 7761 . . . 4 ( 𝐷 ∈ V → 𝐷 ∈ V)
3 mptexg 7242 . . . 4 ( 𝐷 ∈ V → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
41, 2, 33syl 18 . . 3 (𝐷 ∈ DirRel → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
5 unieq 4917 . . . . . 6 (𝑑 = 𝐷 𝑑 = 𝐷)
65unieqd 4919 . . . . 5 (𝑑 = 𝐷 𝑑 = 𝐷)
7 imaeq1 6072 . . . . 5 (𝑑 = 𝐷 → (𝑑 “ {𝑥}) = (𝐷 “ {𝑥}))
86, 7mpteq12dv 5232 . . . 4 (𝑑 = 𝐷 → (𝑥 𝑑 ↦ (𝑑 “ {𝑥})) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
9 df-tail 18643 . . . 4 tail = (𝑑 ∈ DirRel ↦ (𝑥 𝑑 ↦ (𝑑 “ {𝑥})))
108, 9fvmptg 7013 . . 3 ((𝐷 ∈ DirRel ∧ (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) → (tail‘𝐷) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
114, 10mpdan 687 . 2 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
12 tailfval.1 . . . 4 𝑋 = dom 𝐷
13 dirdm 18646 . . . 4 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
1412, 13eqtr2id 2789 . . 3 (𝐷 ∈ DirRel → 𝐷 = 𝑋)
1514mpteq1d 5236 . 2 (𝐷 ∈ DirRel → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
1611, 15eqtrd 2776 1 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  Vcvv 3479  {csn 4625   cuni 4906  cmpt 5224  dom cdm 5684  cima 5687  cfv 6560  DirRelcdir 18640  tailctail 18641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-dir 18642  df-tail 18643
This theorem is referenced by:  tailval  36375  tailf  36377
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