Theorem uspgrupgrushgr 29264

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 29263	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
2		uspgrushgr 29262	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph)
3	1, 2	jca 511	. 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph))
4		eqid 2737	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2737	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	ushgrf 29148	. . . 4 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
7		edgval 29134	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
8		upgredgss 29217	. . . . 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 6744	. . . 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 29237	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
13	12	adantr 480	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
14	11, 13	mpbird 257	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph) → 𝐺 ∈ USPGraph)
15	3, 14	impbii 209	1 ⊢ (𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  {crab 3401   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {csn 4582   class class class wbr 5100  dom cdm 5632  ran crn 5633  –1-1→wf1 6497  ‘cfv 6500   ≤ cle 11179  2c2 12212  ♯chash 14265  Vtxcvtx 29081  iEdgciedg 29082  Edgcedg 29132  USHGraphcushgr 29142  UPGraphcupgr 29165  USPGraphcuspgr 29233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fv 6508  df-edg 29133  df-ushgr 29144  df-upgr 29167  df-uspgr 29235

This theorem is referenced by: (None)