Theorem clwwlkfvOLD 27555

Description: Obsolete version of clwwlkfv 27560 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
clwwlkf1o.d	⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}
clwwlkf1oOLD.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))

Assertion

Ref	Expression
clwwlkfvOLD	⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (𝑊 substr ⟨0, 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 clwwlkfvOLD

Step	Hyp	Ref	Expression
1		oveq1 6977	. 2 ⊢ (𝑡 = 𝑊 → (𝑡 substr ⟨0, 𝑁⟩) = (𝑊 substr ⟨0, 𝑁⟩))
2		clwwlkf1oOLD.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
3		ovex 7002	. 2 ⊢ (𝑊 substr ⟨0, 𝑁⟩) ∈ V
4	1, 2, 3	fvmpt 6589	1 ⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (𝑊 substr ⟨0, 𝑁⟩))

Syntax hints:   → wi 4   = wceq 1507   ∈ wcel 2048  {crab 3086  ⟨cop 4441   ↦ cmpt 5002  ‘cfv 6182  (class class class)co 6970  0cc0 10327  lastSclsw 13715   substr csubstr 13793   WWalksN cwwlksn 27302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-iota 6146  df-fun 6184  df-fv 6190  df-ov 6973

This theorem is referenced by:  clwwlkf1OLD  27556  clwwlkfoOLD  27557