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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 27557 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
2 | uspgrushgr 27556 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph) | |
3 | 1, 2 | jca 512 | . 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph)) |
4 | eqid 2740 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
5 | eqid 2740 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
6 | 4, 5 | ushgrf 27444 | . . . 4 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅})) |
7 | edgval 27430 | . . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
8 | upgredgss 27513 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) | |
9 | 7, 8 | eqsstrrid 3975 | . . . 4 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
10 | f1ssr 6675 | . . . 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 597 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) |
12 | 4, 5 | isuspgr 27533 | . . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
13 | 12 | adantr 481 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})) |
14 | 11, 13 | mpbird 256 | . 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 396 ∈ wcel 2110 {crab 3070 ∖ cdif 3889 ⊆ wss 3892 ∅c0 4262 𝒫 cpw 4539 {csn 4567 class class class wbr 5079 dom cdm 5590 ran crn 5591 –1-1→wf1 6429 ‘cfv 6432 ≤ cle 11021 2c2 12039 ♯chash 14055 Vtxcvtx 27377 iEdgciedg 27378 Edgcedg 27428 USHGraphcushgr 27438 UPGraphcupgr 27461 USPGraphcuspgr 27529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7583 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fv 6440 df-edg 27429 df-ushgr 27440 df-upgr 27463 df-uspgr 27531 |
This theorem is referenced by: (None) |
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