Theorem wlkResOLD 28661

Description: Obsolete version of opabresex2 7414 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 1798	. 2 ⊢ ∀𝑓∀𝑝(𝑓(𝑊‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
3		wksv 28630	. 2 ⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
4		opabbrex 7413	. 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 690	1 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑊‘𝐺)𝑝 ∧ 𝜑)} ∈ V

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1539   ∈ wcel 2106  Vcvv 3446   class class class wbr 5110  {copab 5172  ‘cfv 6501  Walkscwlks 28607

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5261  ax-nul 5268

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-sn 4592  df-pr 4594  df-uni 4871  df-br 5111  df-opab 5173  df-iota 6453  df-fv 6509

This theorem is referenced by: (None)