Theorem uspgrupgrushgr 26072

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 26071	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
2		uspgrushgr 26070	. . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph )
3	1, 2	jca 554	. 2 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ))
4		eqid 2622	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2622	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
6	4, 5	ushgrf 25958	. . . 4 ⊢ (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
7		edgval 25941	. . . . 5 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
8		upgredgss 26027	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
9	7, 8	syl5eqssr 3650	. . . 4 ⊢ (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
10		f1ssr 6107	. . . 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 495	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
12	4, 5	isuspgr 26047	. . . 4 ⊢ (𝐺 ∈ UPGraph → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
13	12	adantr 481	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → (𝐺 ∈ USPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
14	11, 13	mpbird 247	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ) → 𝐺 ∈ USPGraph )
15	3, 14	impbii 199	1 ⊢ (𝐺 ∈ USPGraph ↔ (𝐺 ∈ UPGraph ∧ 𝐺 ∈ USHGraph ))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∈ wcel 1990  {crab 2916   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177   class class class wbr 4653  dom cdm 5114  ran crn 5115  –1-1→wf1 5885  ‘cfv 5888   ≤ cle 10075  2c2 11070  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   USHGraph cushgr 25952   UPGraph cupgr 25975   USPGraph cuspgr 26043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-edg 25940  df-ushgr 25954  df-upgr 25977  df-uspgr 26045

This theorem is referenced by: (None)