Theorem tailfval 32955

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 7232	. . . 4 ⊢ (𝐷 ∈ DirRel → ∪ 𝐷 ∈ V)
2		uniexg 7232	. . . 4 ⊢ (∪ 𝐷 ∈ V → ∪ ∪ 𝐷 ∈ V)
3		mptexg 6756	. . . 4 ⊢ (∪ ∪ 𝐷 ∈ V → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
5		unieq 4679	. . . . . 6 ⊢ (𝑑 = 𝐷 → ∪ 𝑑 = ∪ 𝐷)
6	5	unieqd 4681	. . . . 5 ⊢ (𝑑 = 𝐷 → ∪ ∪ 𝑑 = ∪ ∪ 𝐷)
7		imaeq1 5715	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 “ {𝑥}) = (𝐷 “ {𝑥}))
8	6, 7	mpteq12dv 4969	. . . 4 ⊢ (𝑑 = 𝐷 → (𝑥 ∈ ∪ ∪ 𝑑 ↦ (𝑑 “ {𝑥})) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})))
9		df-tail 17617	. . . 4 ⊢ tail = (𝑑 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑑 ↦ (𝑑 “ {𝑥})))
10	8, 9	fvmptg 6540	. . 3 ⊢ ((𝐷 ∈ DirRel ∧ (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) → (tail‘𝐷) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})))
11	4, 10	mpdan 677	. 2 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})))
12		tailfval.1	. . . 4 ⊢ 𝑋 = dom 𝐷
13		dirdm 17620	. . . 4 ⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪ ∪ 𝐷)
14	12, 13	syl5req 2827	. . 3 ⊢ (𝐷 ∈ DirRel → ∪ ∪ 𝐷 = 𝑋)
15	14	mpteq1d 4973	. 2 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))
16	11, 15	eqtrd 2814	1 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))

Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  Vcvv 3398  {csn 4398  ∪ cuni 4671   ↦ cmpt 4965  dom cdm 5355   “ cima 5358  ‘cfv 6135  DirRelcdir 17614  tailctail 17615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-dir 17616  df-tail 17617

This theorem is referenced by:  tailval  32956  tailf  32958