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) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))

Distinct variable groups:   𝑥,𝐷   𝑥,𝑋

Proof of Theorem tailfval

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	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)
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
5		unieq 4917	. . . . . 6 ⊢ (𝑑 = 𝐷 → ∪ 𝑑 = ∪ 𝐷)
6	5	unieqd 4919	. . . . 5 ⊢ (𝑑 = 𝐷 → ∪ ∪ 𝑑 = ∪ ∪ 𝐷)
7		imaeq1 6072	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 “ {𝑥}) = (𝐷 “ {𝑥}))
8	6, 7	mpteq12dv 5232	. . . 4 ⊢ (𝑑 = 𝐷 → (𝑥 ∈ ∪ ∪ 𝑑 ↦ (𝑑 “ {𝑥})) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})))
9		df-tail 18643	. . . 4 ⊢ tail = (𝑑 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑑 ↦ (𝑑 “ {𝑥})))
10	8, 9	fvmptg 7013	. . 3 ⊢ ((𝐷 ∈ DirRel ∧ (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) → (tail‘𝐷) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})))
11	4, 10	mpdan 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)
14	12, 13	eqtr2id 2789	. . 3 ⊢ (𝐷 ∈ DirRel → ∪ ∪ 𝐷 = 𝑋)
15	14	mpteq1d 5236	. 2 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))
16	11, 15	eqtrd 2776	1 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))

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