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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 4080 | . . . 4 ⊢ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 𝑉 ∖ {∅}) | |
| 2 | difss 4136 | . . . 4 ⊢ (𝒫 𝑉 ∖ {∅}) ⊆ 𝒫 𝑉 | |
| 3 | 1, 2 | sstri 3993 | . . 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 29103 | . . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
| 7 | 6 | ffvelcdmda 7104 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
| 8 | 3, 7 | sselid 3981 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ∈ 𝒫 𝑉) |
| 9 | 8 | elpwid 4609 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 ≤ cle 11296 2c2 12321 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 UPGraphcupgr 29097 |
| 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-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-upgr 29099 |
| This theorem is referenced by: upgrex 29109 isuspgrim0 47872 |
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