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Theorem tailfval 36084
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 7751 . . . 4 (𝐷 ∈ DirRel → 𝐷 ∈ V)
2 uniexg 7751 . . . 4 ( 𝐷 ∈ V → 𝐷 ∈ V)
3 mptexg 7238 . . . 4 ( 𝐷 ∈ V → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
41, 2, 33syl 18 . . 3 (𝐷 ∈ DirRel → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
5 unieq 4924 . . . . . 6 (𝑑 = 𝐷 𝑑 = 𝐷)
65unieqd 4926 . . . . 5 (𝑑 = 𝐷 𝑑 = 𝐷)
7 imaeq1 6064 . . . . 5 (𝑑 = 𝐷 → (𝑑 “ {𝑥}) = (𝐷 “ {𝑥}))
86, 7mpteq12dv 5244 . . . 4 (𝑑 = 𝐷 → (𝑥 𝑑 ↦ (𝑑 “ {𝑥})) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
9 df-tail 18622 . . . 4 tail = (𝑑 ∈ DirRel ↦ (𝑥 𝑑 ↦ (𝑑 “ {𝑥})))
108, 9fvmptg 7007 . . 3 ((𝐷 ∈ DirRel ∧ (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) → (tail‘𝐷) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
114, 10mpdan 685 . 2 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
12 tailfval.1 . . . 4 𝑋 = dom 𝐷
13 dirdm 18625 . . . 4 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
1412, 13eqtr2id 2779 . . 3 (𝐷 ∈ DirRel → 𝐷 = 𝑋)
1514mpteq1d 5248 . 2 (𝐷 ∈ DirRel → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
1611, 15eqtrd 2766 1 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3462  {csn 4633   cuni 4913  cmpt 5236  dom cdm 5682  cima 5685  cfv 6554  DirRelcdir 18619  tailctail 18620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-dir 18621  df-tail 18622
This theorem is referenced by:  tailval  36085  tailf  36087
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