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

Theorem wwlksswrd 26216
 Description: Walks (represented by words) are words. (Contributed by Alexander van der Vekens, 17-Jul-2018.)
Assertion
Ref Expression
wwlksswrd (𝑉 WWalks 𝐸) ⊆ Word 𝑉

Proof of Theorem wwlksswrd
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 wwlkprop 26213 . . 3 (𝑤 ∈ (𝑉 WWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑤 ∈ Word 𝑉))
21simp3d 1068 . 2 (𝑤 ∈ (𝑉 WWalks 𝐸) → 𝑤 ∈ Word 𝑉)
32ssriv 3572 1 (𝑉 WWalks 𝐸) ⊆ Word 𝑉
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  (class class class)co 6549  Word cword 13146   WWalks cwwlk 26205 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-8 1979  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-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  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-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  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-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-pm 7747  df-neg 10148  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-word 13154  df-wwlk 26207 This theorem is referenced by:  disjxwwlkn  26273
 Copyright terms: Public domain W3C validator