Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > uspgrushgr | Structured version Visualization version GIF version |
Description: A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.) |
Ref | Expression |
---|---|
uspgrushgr | ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | eqid 2610 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | isuspgr 40382 | . . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})) |
4 | ssrab2 3650 | . . . . 5 ⊢ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}) | |
5 | f1ss 6019 | . . . . 5 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ∧ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅})) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})) | |
6 | 4, 5 | mpan2 703 | . . . 4 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})) |
7 | 3, 6 | syl6bi 242 | . . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
8 | 1, 2 | isushgr 25727 | . . 3 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
9 | 7, 8 | sylibrd 248 | . 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph )) |
10 | 9 | pm2.43i 50 | 1 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 {crab 2900 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 dom cdm 5038 –1-1→wf1 5801 ‘cfv 5804 ≤ cle 9954 2c2 10947 #chash 12979 Vtxcvtx 25673 iEdgciedg 25674 USHGraph cushgr 25723 USPGraph cuspgr 40378 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fv 5812 df-ushgr 25725 df-uspgr 40380 |
This theorem is referenced by: uspgrupgrushgr 40407 usgredgedga 40457 vtxdusgrfvedg 40706 1loopgrvd2 40718 |
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