Theorem clwwlkfv 30082

Description: Lemma 2 for clwwlkf1o 30085: 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 7457	. 2 ⊢ (𝑡 = 𝑊 → (𝑡 prefix 𝑁) = (𝑊 prefix 𝑁))
2		clwwlkf1o.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))
3		ovex 7483	. 2 ⊢ (𝑊 prefix 𝑁) ∈ V
4	1, 2, 3	fvmpt 7031	1 ⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (𝑊 prefix 𝑁))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {crab 3443   ↦ cmpt 5249  ‘cfv 6575  (class class class)co 7450  0cc0 11186  lastSclsw 14612   prefix cpfx 14720   WWalksN cwwlksn 29861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6527  df-fun 6577  df-fv 6583  df-ov 7453

This theorem is referenced by:  clwwlkf1  30083  clwwlkfo  30084