Theorem wlkResOLD 29686

Description: Obsolete version of opabresex2 7502 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 1794	. 2 ⊢ ∀𝑓∀𝑝(𝑓(𝑊‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝)
3		wksv 29655	. 2 ⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
4		opabbrex 7501	. 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 395  ∀wal 1535   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  {copab 5228  ‘cfv 6573  Walkscwlks 29632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-sn 4649  df-pr 4651  df-uni 4932  df-br 5167  df-opab 5229  df-iota 6525  df-fv 6581

This theorem is referenced by: (None)