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Theorem clwwlkfv 27843
 Description: Lemma 2 for clwwlkf1o 27846: the value of function F. (Contributed by Alexander van der Vekens, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
Hypotheses
Ref Expression
clwwlkf1o.d 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}
clwwlkf1o.f 𝐹 = (𝑡𝐷 ↦ (𝑡 prefix 𝑁))
Assertion
Ref Expression
clwwlkfv (𝑊𝐷 → (𝐹𝑊) = (𝑊 prefix 𝑁))
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑡,𝐷   𝑡,𝐺,𝑤   𝑡,𝑁   𝑡,𝑊
Allowed substitution hints:   𝐷(𝑤)   𝐹(𝑤,𝑡)   𝑊(𝑤)

Proof of Theorem clwwlkfv
StepHypRef Expression
1 oveq1 7143 . 2 (𝑡 = 𝑊 → (𝑡 prefix 𝑁) = (𝑊 prefix 𝑁))
2 clwwlkf1o.f . 2 𝐹 = (𝑡𝐷 ↦ (𝑡 prefix 𝑁))
3 ovex 7169 . 2 (𝑊 prefix 𝑁) ∈ V
41, 2, 3fvmpt 6746 1 (𝑊𝐷 → (𝐹𝑊) = (𝑊 prefix 𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110   ↦ cmpt 5111  ‘cfv 6325  (class class class)co 7136  0cc0 10529  lastSclsw 13908   prefix cpfx 14026   WWalksN cwwlksn 27622 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333  df-ov 7139 This theorem is referenced by:  clwwlkf1  27844  clwwlkfo  27845
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