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| Mirrors > Home > MPE Home > Th. List > wksv | Structured version Visualization version GIF version | ||
| Description: The class of walks is a set. (Contributed by AV, 15-Jan-2021.) (Proof shortened by SN, 11-Dec-2024.) | 
| Ref | Expression | 
|---|---|
| wksv | ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvex 6919 | . 2 ⊢ (Walks‘𝐺) ∈ V | |
| 2 | opabss 5207 | . 2 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ⊆ (Walks‘𝐺) | |
| 3 | 1, 2 | ssexi 5322 | 1 ⊢ {〈𝑓, 𝑝〉 ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 {copab 5205 ‘cfv 6561 Walkscwlks 29614 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 df-br 5144 df-opab 5206 df-iota 6514 df-fv 6569 | 
| This theorem is referenced by: wlkResOLD 29668 wksonproplemOLD 29723 | 
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