Theorem clwwlkfv 29992

Description: Lemma 2 for clwwlkf1o 29995: 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 7356	. 2 ⊢ (𝑡 = 𝑊 → (𝑡 prefix 𝑁) = (𝑊 prefix 𝑁))
2		clwwlkf1o.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))
3		ovex 7382	. 2 ⊢ (𝑊 prefix 𝑁) ∈ V
4	1, 2, 3	fvmpt 6930	1 ⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (𝑊 prefix 𝑁))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3394   ↦ cmpt 5173  ‘cfv 6482  (class class class)co 7349  0cc0 11009  lastSclsw 14469   prefix cpfx 14577   WWalksN cwwlksn 29771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352

This theorem is referenced by:  clwwlkf1  29993  clwwlkfo  29994