Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwwlkfv Structured version   Visualization version   GIF version

Theorem clwwlkfv 26323
 Description: Lemma 2 for clwwlkbij 26327: the value of function F. (Contributed by Alexander van der Vekens, 28-Sep-2018.)
Hypotheses
Ref Expression
clwwlkbij.d 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
clwwlkbij.f 𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
Assertion
Ref Expression
clwwlkfv (𝑊𝐷 → (𝐹𝑊) = (𝑊 substr ⟨0, 𝑁⟩))
Distinct variable groups:   𝑤,𝐸   𝑤,𝑁   𝑤,𝑉   𝑡,𝐷   𝑡,𝐸,𝑤   𝑡,𝑁   𝑡,𝑉   𝑡,𝑊
Allowed substitution hints:   𝐷(𝑤)   𝐹(𝑤,𝑡)   𝑊(𝑤)

Proof of Theorem clwwlkfv
StepHypRef Expression
1 oveq1 6556 . 2 (𝑡 = 𝑊 → (𝑡 substr ⟨0, 𝑁⟩) = (𝑊 substr ⟨0, 𝑁⟩))
2 clwwlkbij.f . 2 𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
3 ovex 6577 . 2 (𝑊 substr ⟨0, 𝑁⟩) ∈ V
41, 2, 3fvmpt 6191 1 (𝑊𝐷 → (𝐹𝑊) = (𝑊 substr ⟨0, 𝑁⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900  ⟨cop 4131   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  0cc0 9815   lastS clsw 13147   substr csubstr 13150   WWalksN cwwlkn 26206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552 This theorem is referenced by:  clwwlkf1  26324  clwwlkfo  26325
 Copyright terms: Public domain W3C validator