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Theorem tailfval 32594
 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 7072 . . . 4 (𝐷 ∈ DirRel → 𝐷 ∈ V)
2 uniexg 7072 . . . 4 ( 𝐷 ∈ V → 𝐷 ∈ V)
3 mptexg 6600 . . . 4 ( 𝐷 ∈ V → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
41, 2, 33syl 18 . . 3 (𝐷 ∈ DirRel → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V)
5 unieq 4552 . . . . . 6 (𝑑 = 𝐷 𝑑 = 𝐷)
65unieqd 4554 . . . . 5 (𝑑 = 𝐷 𝑑 = 𝐷)
7 imaeq1 5571 . . . . 5 (𝑑 = 𝐷 → (𝑑 “ {𝑥}) = (𝐷 “ {𝑥}))
86, 7mpteq12dv 4841 . . . 4 (𝑑 = 𝐷 → (𝑥 𝑑 ↦ (𝑑 “ {𝑥})) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
9 df-tail 17353 . . . 4 tail = (𝑑 ∈ DirRel ↦ (𝑥 𝑑 ↦ (𝑑 “ {𝑥})))
108, 9fvmptg 6394 . . 3 ((𝐷 ∈ DirRel ∧ (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) → (tail‘𝐷) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
114, 10mpdan 705 . 2 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 𝐷 ↦ (𝐷 “ {𝑥})))
12 tailfval.1 . . . 4 𝑋 = dom 𝐷
13 dirdm 17356 . . . 4 (𝐷 ∈ DirRel → dom 𝐷 = 𝐷)
1412, 13syl5req 2771 . . 3 (𝐷 ∈ DirRel → 𝐷 = 𝑋)
1514mpteq1d 4846 . 2 (𝐷 ∈ DirRel → (𝑥 𝐷 ↦ (𝐷 “ {𝑥})) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
1611, 15eqtrd 2758 1 (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥𝑋 ↦ (𝐷 “ {𝑥})))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1596   ∈ wcel 2103  Vcvv 3304  {csn 4285  ∪ cuni 4544   ↦ cmpt 4837  dom cdm 5218   “ cima 5221  ‘cfv 6001  DirRelcdir 17350  tailctail 17351 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pr 5011  ax-un 7066 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-dir 17352  df-tail 17353 This theorem is referenced by:  tailval  32595  tailf  32597
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