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Theorem tailfval 31371
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 6729 . . . 4 (𝐷 ∈ DirRel → 𝐷 ∈ V)
2 uniexg 6729 . . . 4 ( 𝐷 ∈ V → 𝐷 ∈ V)
3 mptexg 6266 . . . 4 ( 𝐷 ∈ V → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
41, 2, 33syl 18 . . 3 (𝐷 ∈ DirRel → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
5 unieq 4278 . . . . . 6 (𝑑 = 𝐷 𝑑 = 𝐷)
65unieqd 4280 . . . . 5 (𝑑 = 𝐷 𝑑 = 𝐷)
7 imaeq1 5271 . . . . 5 (𝑑 = 𝐷 → (𝑑 “ {𝑥}) = (𝐷 “ {𝑥}))
86, 7mpteq12dv 4561 . . . 4 (𝑑 = 𝐷 → (𝑥 𝑑 ↦ (𝑑 “ {𝑥})) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
9 df-tail 16946 . . . 4 tail = (𝑑 ∈ DirRel ↦ (𝑥 𝑑 ↦ (𝑑 “ {𝑥})))
108, 9fvmptg 6073 . . 3 ((𝐷 ∈ DirRel ∧ (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) → (tail‘𝐷) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
114, 10mpdan 698 . 2 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
12 tailfval.1 . . . 4 𝑋 = dom 𝐷
13 dirdm 16949 . . . 4 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
1412, 13syl5req 2561 . . 3 (𝐷 ∈ DirRel → 𝐷 = 𝑋)
1514mpteq1d 4564 . 2 (𝐷 ∈ DirRel → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
1611, 15eqtrd 2548 1 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1938  Vcvv 3077  {csn 4028   cuni 4270  cmpt 4541  dom cdm 4932  cima 4935  cfv 5689  DirRelcdir 16943  tailctail 16944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-dir 16945  df-tail 16946
This theorem is referenced by:  tailval  31372  tailf  31374
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