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Mirrors > Home > MPE Home > Th. List > wlkResOLD | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
wlkResOLD.1 | ⊢ (𝑓(𝑊‘𝐺)𝑝 → 𝑓(Walks‘𝐺)𝑝) |
Ref | Expression |
---|---|
wlkResOLD | ⊢ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑊‘𝐺)𝑝 ∧ 𝜑)} ∈ V |
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 28396 | . 2 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V | |
4 | opabbrex 7403 | . 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 |
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) |
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