Theorem uspgrupgrushgr 26956

Description: A graph is a simple pseudograph iff it is a pseudograph and a simple hypergraph. (Contributed by AV, 30-Nov-2020.)

Assertion

Ref	Expression
uspgrupgrushgr	⊢ (𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph))

Proof of Theorem uspgrupgrushgr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uspgrupgr 26955	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
2		uspgrushgr 26954	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
3	1, 2	jca 514	. 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph))
4		eqid 2821	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2821	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	ushgrf 26842	. . . 4 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
7		edgval 26828	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
8		upgredgss 26911	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
9	7, 8	eqsstrrid 4015	. . . 4 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
10		f1ssr 6575	. . . 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 598	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
12	4, 5	isuspgr 26931	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
13	12	adantr 483	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
14	11, 13	mpbird 259	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → 𝐺 ∈ USPGraph)
15	3, 14	impbii 211	1 ⊢ (𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  {crab 3142   ∖ cdif 3932   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  {csn 4560   class class class wbr 5058  dom cdm 5549  ran crn 5550  –1-1→wf1 6346  ‘cfv 6349   ≤ cle 10670  2c2 11686  ♯chash 13684  Vtxcvtx 26775  iEdgciedg 26776  Edgcedg 26826  USHGraphcushgr 26836  UPGraphcupgr 26859  USPGraphcuspgr 26927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fv 6357  df-edg 26827  df-ushgr 26838  df-upgr 26861  df-uspgr 26929

This theorem is referenced by: (None)