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Theorem uspgrupgrushgr 40407
 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.
StepHypRef Expression
1 uspgrupgr 40406 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
2 uspgrushgr 40405 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph )
31, 2jca 553 . 2 (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ))
4 eqid 2610 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2610 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5ushgrf 25729 . . . 4 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
7 edgaval 25794 . . . . 5 (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
8 upgredgss 25806 . . . . 5 (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
97, 8eqsstr3d 3603 . . . 4 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
10 f1ssr 6020 . . . 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})
116, 9, 10syl2anr 494 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
124, 5isuspgr 40382 . . . 4 (𝐺 ∈ UPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
1312adantr 480 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
1411, 13mpbird 246 . 2 ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → 𝐺 ∈ USPGraph )
153, 14impbii 198 1 (𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ran crn 5039  –1-1→wf1 5801  ‘cfv 5804   ≤ cle 9954  2c2 10947  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   USHGraph cushgr 25723   UPGraph cupgr 25747  Edgcedga 25792   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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 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-mo 2463  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-csb 3500  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-mpt 4645  df-id 4953  df-xp 5044  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-upgr 25749  df-edga 25793  df-uspgr 40380 This theorem is referenced by: (None)
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