Theorem wksv 29451

Description: The class of walks is a set. (Contributed by AV, 15-Jan-2021.) (Proof shortened by SN, 11-Dec-2024.)

Assertion

Ref	Expression
wksv	⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V

Distinct variable group:   𝑓,𝐺,𝑝

Proof of Theorem wksv

Step	Hyp	Ref	Expression
1		fvex 6913	. 2 ⊢ (Walks‘𝐺) ∈ V
2		opabss 5214	. 2 ⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ⊆ (Walks‘𝐺)
3	1, 2	ssexi 5324	1 ⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V

Syntax hints:   ∈ wcel 2098  Vcvv 3471   class class class wbr 5150  {copab 5212  ‘cfv 6551  Walkscwlks 29428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-sn 4631  df-pr 4633  df-uni 4911  df-br 5151  df-opab 5213  df-iota 6503  df-fv 6559

This theorem is referenced by:  wlkResOLD  29482  wksonproplemOLD  29537