Theorem wlkResOLD 29174

Description: Obsolete version of opabresex2 7463 as of 13-Dec-2024. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 30-Dec-2020.) (Proof shortened by AV, 15-Jan-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
wlkResOLD.1	⊢ (𝑓(𝑊‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)

Assertion

Ref	Expression
wlkResOLD	⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑊‘𝐺)𝑝 ∧ 𝜑)} ∈ V

Distinct variable group:   𝑓,𝐺,𝑝

Allowed substitution hints:   𝜑(𝑓,𝑝)   𝑊(𝑓,𝑝)

Proof of Theorem wlkResOLD

Step	Hyp	Ref	Expression
1		wlkResOLD.1	. . 3 ⊢ (𝑓(𝑊‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
2	1	gen2 1796	. 2 ⊢ ∀𝑓∀𝑝(𝑓(𝑊‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
3		wksv 29143	. 2 ⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
4		opabbrex 7462	. 2 ⊢ ((∀𝑓∀𝑝(𝑓(𝑊‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝) ∧ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑊‘𝐺)𝑝 ∧ 𝜑)} ∈ V)
5	2, 3, 4	mp2an 688	1 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑊‘𝐺)𝑝 ∧ 𝜑)} ∈ V

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1537   ∈ wcel 2104  Vcvv 3472   class class class wbr 5147  {copab 5209  ‘cfv 6542  Walkscwlks 29120

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-sn 4628  df-pr 4630  df-uni 4908  df-br 5148  df-opab 5210  df-iota 6494  df-fv 6550

This theorem is referenced by: (None)