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Mirrors > Home > MPE Home > Th. List > upgrss | Structured version Visualization version GIF version |
Description: An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 29-Nov-2020.) |
Ref | Expression |
---|---|
isupgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isupgr.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
upgrss | ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3967 | . . . 4 ⊢ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 𝑉 ∖ {∅}) | |
2 | difss 4020 | . . . 4 ⊢ (𝒫 𝑉 ∖ {∅}) ⊆ 𝒫 𝑉 | |
3 | 1, 2 | sstri 3884 | . . 3 ⊢ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ 𝒫 𝑉 |
4 | isupgr.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | isupgr.e | . . . . 5 ⊢ 𝐸 = (iEdg‘𝐺) | |
6 | 4, 5 | upgrf 27023 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
7 | 6 | ffvelrnda 6855 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
8 | 3, 7 | sseldi 3873 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ 𝒫 𝑉) |
9 | 8 | elpwid 4496 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 {crab 3057 ∖ cdif 3838 ⊆ wss 3841 ∅c0 4209 𝒫 cpw 4485 {csn 4513 class class class wbr 5027 dom cdm 5519 ‘cfv 6333 ≤ cle 10747 2c2 11764 ♯chash 13775 Vtxcvtx 26933 iEdgciedg 26934 UPGraphcupgr 27017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fv 6341 df-upgr 27019 |
This theorem is referenced by: upgrex 27029 |
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