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Theorem uspgrupgrushgr 26889
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 26888 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
2 uspgrushgr 26887 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
31, 2jca 512 . 2 (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph))
4 eqid 2818 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2818 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5ushgrf 26775 . . . 4 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
7 edgval 26761 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
8 upgredgss 26844 . . . . 5 (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
97, 8eqsstrrid 4013 . . . 4 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
10 f1ssr 6574 . . . 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 596 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
124, 5isuspgr 26864 . . . 4 (𝐺 ∈ UPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1312adantr 481 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1411, 13mpbird 258 . 2 ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → 𝐺 ∈ USPGraph)
153, 14impbii 210 1 (𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2105  {crab 3139  cdif 3930  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557   class class class wbr 5057  dom cdm 5548  ran crn 5549  1-1wf1 6345  cfv 6348  cle 10664  2c2 11680  chash 13678  Vtxcvtx 26708  iEdgciedg 26709  Edgcedg 26759  USHGraphcushgr 26769  UPGraphcupgr 26792  USPGraphcuspgr 26860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fv 6356  df-edg 26760  df-ushgr 26771  df-upgr 26794  df-uspgr 26862
This theorem is referenced by: (None)
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