Theorem clwwlkfv 28313

Description: Lemma 2 for clwwlkf1o 28316: 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 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))

Assertion

Ref	Expression
clwwlkfv	⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (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

Step	Hyp	Ref	Expression
1		oveq1 7262	. 2 ⊢ (𝑡 = 𝑊 → (𝑡 prefix 𝑁) = (𝑊 prefix 𝑁))
2		clwwlkf1o.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))
3		ovex 7288	. 2 ⊢ (𝑊 prefix 𝑁) ∈ V
4	1, 2, 3	fvmpt 6857	1 ⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (𝑊 prefix 𝑁))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {crab 3067   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255  0cc0 10802  lastSclsw 14193   prefix cpfx 14311   WWalksN cwwlksn 28092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258

This theorem is referenced by:  clwwlkf1  28314  clwwlkfo  28315