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Mirrors > Home > MPE Home > Th. List > uhgrss | Structured version Visualization version GIF version |
Description: An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.) |
Ref | Expression |
---|---|
uhgrf.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgrf.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
uhgrss | ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrf.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | uhgrf.e | . . . . 5 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | uhgrf 27899 | . . . 4 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
4 | 3 | ffvelcdmda 7031 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ (𝒫 𝑉 ∖ {∅})) |
5 | 4 | eldifad 3920 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ 𝒫 𝑉) |
6 | 5 | elpwid 4567 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3905 ⊆ wss 3908 ∅c0 4280 𝒫 cpw 4558 {csn 4584 dom cdm 5631 ‘cfv 6493 Vtxcvtx 27833 iEdgciedg 27834 UHGraphcuhgr 27893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-uhgr 27895 |
This theorem is referenced by: lpvtx 27905 umgredgprv 27944 uhgrspansubgrlem 28124 uhgrspan1 28137 isomgreqve 45949 isomgrsym 45960 ushrisomgr 45965 |
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