Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 27153 | . . . 4 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
4 | 3 | ffvelrnda 6904 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ (𝒫 𝑉 ∖ {∅})) |
5 | 4 | eldifad 3878 | . 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ 𝒫 𝑉) |
6 | 5 | elpwid 4524 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∖ cdif 3863 ⊆ wss 3866 ∅c0 4237 𝒫 cpw 4513 {csn 4541 dom cdm 5551 ‘cfv 6380 Vtxcvtx 27087 iEdgciedg 27088 UHGraphcuhgr 27147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fv 6388 df-uhgr 27149 |
This theorem is referenced by: lpvtx 27159 umgredgprv 27198 uhgrspansubgrlem 27378 uhgrspan1 27391 isomgreqve 44950 isomgrsym 44961 ushrisomgr 44966 |
Copyright terms: Public domain | W3C validator |