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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) = (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 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)
41, 2, 33syl 18 . . 3 (𝐷 ∈ DirRel → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
5 unieq 4679 . . . . . 6 (𝑑 = 𝐷 𝑑 = 𝐷)
65unieqd 4681 . . . . 5 (𝑑 = 𝐷 𝑑 = 𝐷)
7 imaeq1 5715 . . . . 5 (𝑑 = 𝐷 → (𝑑 “ {𝑥}) = (𝐷 “ {𝑥}))
86, 7mpteq12dv 4969 . . . 4 (𝑑 = 𝐷 → (𝑥 𝑑 ↦ (𝑑 “ {𝑥})) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
9 df-tail 17617 . . . 4 tail = (𝑑 ∈ DirRel ↦ (𝑥 𝑑 ↦ (𝑑 “ {𝑥})))
108, 9fvmptg 6540 . . 3 ((𝐷 ∈ DirRel ∧ (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) → (tail‘𝐷) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
114, 10mpdan 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)
1412, 13syl5req 2827 . . 3 (𝐷 ∈ DirRel → 𝐷 = 𝑋)
1514mpteq1d 4973 . 2 (𝐷 ∈ DirRel → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
1611, 15eqtrd 2814 1 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
Colors of variables: wff setvar class
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
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