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Mirrors > Home > MPE Home > Th. List > upgredgss | Structured version Visualization version GIF version |
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 27805 | . 2 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
2 | eqid 2738 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | eqid 2738 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
4 | 2, 3 | upgrf 27842 | . . 3 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
5 | 4 | frnd 6672 | . 2 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
6 | 1, 5 | eqsstrid 3991 | 1 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 {crab 3406 ∖ cdif 3906 ⊆ wss 3909 ∅c0 4281 𝒫 cpw 4559 {csn 4585 class class class wbr 5104 dom cdm 5631 ran crn 5632 ‘cfv 6492 ≤ cle 11124 2c2 12142 ♯chash 14159 Vtxcvtx 27752 iEdgciedg 27753 Edgcedg 27803 UPGraphcupgr 27836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-edg 27804 df-upgr 27838 |
This theorem is referenced by: uspgrupgrushgr 27933 upgredgssspr 45836 |
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