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Mirrors > Home > MPE Home > Th. List > uspgrupgrushgr | Structured version Visualization version GIF version |
Description: A graph is a simple pseudograph iff it is a pseudograph and a simple hypergraph. (Contributed by AV, 30-Nov-2020.) |
Ref | Expression |
---|---|
uspgrupgrushgr | ⊢ (𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrupgr 28978 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
2 | uspgrushgr 28977 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph) | |
3 | 1, 2 | jca 511 | . 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph)) |
4 | eqid 2727 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
5 | eqid 2727 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
6 | 4, 5 | ushgrf 28863 | . . . 4 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})) |
7 | edgval 28849 | . . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
8 | upgredgss 28932 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
9 | 7, 8 | eqsstrrid 4027 | . . . 4 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
10 | f1ssr 6794 | . . . 4 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
11 | 6, 9, 10 | syl2anr 596 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
12 | 4, 5 | isuspgr 28952 | . . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
13 | 12 | adantr 480 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
14 | 11, 13 | mpbird 257 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → 𝐺 ∈ USPGraph) |
15 | 3, 14 | impbii 208 | 1 ⊢ (𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2099 {crab 3427 ∖ cdif 3941 ⊆ wss 3944 ∅c0 4318 𝒫 cpw 4598 {csn 4624 class class class wbr 5142 dom cdm 5672 ran crn 5673 –1-1→wf1 6539 ‘cfv 6542 ≤ cle 11271 2c2 12289 ♯chash 14313 Vtxcvtx 28796 iEdgciedg 28797 Edgcedg 28847 USHGraphcushgr 28857 UPGraphcupgr 28880 USPGraphcuspgr 28948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fv 6550 df-edg 28848 df-ushgr 28859 df-upgr 28882 df-uspgr 28950 |
This theorem is referenced by: (None) |
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