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| Description: The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 29-Nov-2020.) | 
| Ref | Expression | 
|---|---|
| upgredgss | ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | edgval 29066 | . 2 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
| 2 | eqid 2737 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 3 | eqid 2737 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 4 | 2, 3 | upgrf 29103 | . . 3 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | 
| 5 | 4 | frnd 6744 | . 2 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | 
| 6 | 1, 5 | eqsstrid 4022 | 1 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 {crab 3436 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 class class class wbr 5143 dom cdm 5685 ran crn 5686 ‘cfv 6561 ≤ cle 11296 2c2 12321 ♯chash 14369 Vtxcvtx 29013 iEdgciedg 29014 Edgcedg 29064 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-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| 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-nfc 2892 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-in 3958 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-mpt 5226 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-edg 29065 df-upgr 29099 | 
| This theorem is referenced by: uspgrupgrushgr 29196 upgredgssspr 48059 | 
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