Theorem wlkResOLD 29668

Description: Obsolete version of opabresex2 7485 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 29637	. 2 ⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
4		opabbrex 7484	. 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 692	1 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑊‘𝐺)𝑝 ∧ 𝜑)} ∈ V

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   ∈ wcel 2108  Vcvv 3480   class class class wbr 5143  {copab 5205  ‘cfv 6561  Walkscwlks 29614

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569

This theorem is referenced by: (None)