Theorem clwwlkfv 30118

Description: Lemma 2 for clwwlkf1o 30121: 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 7374	. 2 ⊢ (𝑡 = 𝑊 → (𝑡 prefix 𝑁) = (𝑊 prefix 𝑁))
2		clwwlkf1o.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))
3		ovex 7400	. 2 ⊢ (𝑊 prefix 𝑁) ∈ V
4	1, 2, 3	fvmpt 6947	1 ⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (𝑊 prefix 𝑁))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3389   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367  0cc0 11038  lastSclsw 14524   prefix cpfx 14633   WWalksN cwwlksn 29894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370

This theorem is referenced by:  clwwlkf1  30119  clwwlkfo  30120