MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlkResOLD Structured version   Visualization version   GIF version

Theorem wlkResOLD 28427
Description: Obsolete version of opabresex2 7404 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(𝑊𝐺)𝑝𝑓(Walks‘𝐺)𝑝)
Assertion
Ref Expression
wlkResOLD {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑊𝐺)𝑝𝜑)} ∈ V
Distinct variable group:   𝑓,𝐺,𝑝
Allowed substitution hints:   𝜑(𝑓,𝑝)   𝑊(𝑓,𝑝)

Proof of Theorem wlkResOLD
StepHypRef Expression
1 wlkResOLD.1 . . 3 (𝑓(𝑊𝐺)𝑝𝑓(Walks‘𝐺)𝑝)
21gen2 1799 . 2 𝑓𝑝(𝑓(𝑊𝐺)𝑝𝑓(Walks‘𝐺)𝑝)
3 wksv 28396 . 2 {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
4 opabbrex 7403 . 2 ((∀𝑓𝑝(𝑓(𝑊𝐺)𝑝𝑓(Walks‘𝐺)𝑝) ∧ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑊𝐺)𝑝𝜑)} ∈ V)
52, 3, 4mp2an 691 1 {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑊𝐺)𝑝𝜑)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wcel 2107  Vcvv 3444   class class class wbr 5104  {copab 5166  cfv 6494  Walkscwlks 28373
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 2709  ax-sep 5255  ax-nul 5262
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 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-sn 4586  df-pr 4588  df-uni 4865  df-br 5105  df-opab 5167  df-iota 6446  df-fv 6502
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator