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Theorem upgredgss 29002

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.)

**Assertion**
Ref	Expression
upgredgss	⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Distinct variable group: 𝑥,𝐺

**Proof of Theorem upgredgss**
Step	Hyp	Ref	Expression
1		edgval 28919	. 2 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2		eqid 2725	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2725	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	upgrf 28956	. . 3 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
5	4	frnd 6729	. 2 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
6	1, 5	eqsstrid 4026	1 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2098 {crab 3419 ∖ cdif 3942 ⊆ wss 3945 ∅c0 4323 𝒫 cpw 4603 {csn 4629 class class class wbr 5148 dom cdm 5677 ran crn 5678 ‘cfv 6547 ≤ cle 11279 2c2 12297 ♯chash 14322 Vtxcvtx 28866 iEdgciedg 28867 Edgcedg 28917 UPGraphcupgr 28950

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5299 ax-nul 5306 ax-pr 5428 ax-un 7739

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-fv 6555 df-edg 28918 df-upgr 28952

This theorem is referenced by: uspgrupgrushgr 29049 upgredgssspr 47333

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