Theorem wlkResOLD 28907

Description: Obsolete version of opabresex2 7461 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 1799	. 2 ⊢ ∀𝑓∀𝑝(𝑓(𝑊‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
3		wksv 28876	. 2 ⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
4		opabbrex 7460	. 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 691	1 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑊‘𝐺)𝑝 ∧ 𝜑)} ∈ V

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1540   ∈ wcel 2107  Vcvv 3475   class class class wbr 5149  {copab 5211  ‘cfv 6544  Walkscwlks 28853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-pr 4632  df-uni 4910  df-br 5150  df-opab 5212  df-iota 6496  df-fv 6552

This theorem is referenced by: (None)