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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 4058 | . . . 4 ⊢ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 𝑉 ∖ {∅}) | |
2 | difss 4110 | . . . 4 ⊢ (𝒫 𝑉 ∖ {∅}) ⊆ 𝒫 𝑉 | |
3 | 1, 2 | sstri 3978 | . . 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 26873 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
7 | 6 | ffvelrnda 6853 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
8 | 3, 7 | sseldi 3967 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ 𝒫 𝑉) |
9 | 8 | elpwid 4552 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 ∖ cdif 3935 ⊆ wss 3938 ∅c0 4293 𝒫 cpw 4541 {csn 4569 class class class wbr 5068 dom cdm 5557 ‘cfv 6357 ≤ cle 10678 2c2 11695 ♯chash 13693 Vtxcvtx 26783 iEdgciedg 26784 UPGraphcupgr 26867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-upgr 26869 |
This theorem is referenced by: upgrex 26879 |
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