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Theorem uspgrupgrushgr 26294
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 26293 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
2 uspgrushgr 26292 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
31, 2jca 501 . 2 (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph))
4 eqid 2771 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2771 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5ushgrf 26179 . . . 4 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
7 edgval 26162 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
8 upgredgss 26248 . . . . 5 (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
97, 8syl5eqssr 3799 . . . 4 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
10 f1ssr 6248 . . . 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 584 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
124, 5isuspgr 26269 . . . 4 (𝐺 ∈ UPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1312adantr 466 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1411, 13mpbird 247 . 2 ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → 𝐺 ∈ USPGraph)
153, 14impbii 199 1 (𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wcel 2145  {crab 3065  cdif 3720  wss 3723  c0 4063  𝒫 cpw 4298  {csn 4317   class class class wbr 4787  dom cdm 5250  ran crn 5251  1-1wf1 6027  cfv 6030  cle 10281  2c2 11276  chash 13321  Vtxcvtx 26095  iEdgciedg 26096  Edgcedg 26160  USHGraphcushgr 26173  UPGraphcupgr 26196  USPGraphcuspgr 26265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fv 6038  df-edg 26161  df-ushgr 26175  df-upgr 26198  df-uspgr 26267
This theorem is referenced by: (None)
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