Theorem tailfval 36363

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 7667	. . . 4 ⊢ (𝐷 ∈ DirRel → ∪ 𝐷 ∈ V)
2		uniexg 7667	. . . 4 ⊢ (∪ 𝐷 ∈ V → ∪ ∪ 𝐷 ∈ V)
3		mptexg 7149	. . . 4 ⊢ (∪ ∪ 𝐷 ∈ V → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
5		unieq 4867	. . . . . 6 ⊢ (𝑑 = 𝐷 → ∪ 𝑑 = ∪ 𝐷)
6	5	unieqd 4869	. . . . 5 ⊢ (𝑑 = 𝐷 → ∪ ∪ 𝑑 = ∪ ∪ 𝐷)
7		imaeq1 6000	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 “ {𝑥}) = (𝐷 “ {𝑥}))
8	6, 7	mpteq12dv 5175	. . . 4 ⊢ (𝑑 = 𝐷 → (𝑥 ∈ ∪ ∪ 𝑑 ↦ (𝑑 “ {𝑥})) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})))
9		df-tail 18490	. . . 4 ⊢ tail = (𝑑 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑑 ↦ (𝑑 “ {𝑥})))
10	8, 9	fvmptg 6921	. . 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 18493	. . . 4 ⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪ ∪ 𝐷)
14	12, 13	eqtr2id 2777	. . 3 ⊢ (𝐷 ∈ DirRel → ∪ ∪ 𝐷 = 𝑋)
15	14	mpteq1d 5178	. 2 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))
16	11, 15	eqtrd 2764	1 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3433  {csn 4573  ∪ cuni 4856   ↦ cmpt 5169  dom cdm 5613   “ cima 5616  ‘cfv 6476  DirRelcdir 18487  tailctail 18488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5214  ax-sep 5231  ax-nul 5241  ax-pr 5367  ax-un 7662

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-dir 18489  df-tail 18490

This theorem is referenced by:  tailval  36364  tailf  36366