Theorem usgruspgrben 15984

Description: A class is a simple graph iff it is a simple pseudograph without loops. (Contributed by AV, 18-Oct-2020.)

Assertion

Ref	Expression
usgruspgrben	⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o))

Distinct variable group:   𝑒,𝐺

Proof of Theorem usgruspgrben

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgruspgr 15981	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2		edgusgren 15961	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≈ 2o))
3	2	simprd 114	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ≈ 2o)
4	3	ralrimiva 2603	. . 3 ⊢ (𝐺 ∈ USGraph → ∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o)
5	1, 4	jca 306	. 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o))
6		edgvalg 15860	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
7	6	raleqdv 2734	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o ↔ ∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o))
8		eqid 2229	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2229	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
10	8, 9	uspgrfen 15957	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
11		f1rn 5532	. . . . . . . . 9 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
12		ssel2 3219	. . . . . . . . . . . . . . 15 ⊢ ((ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} ∧ 𝑦 ∈ ran (iEdg‘𝐺)) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
13	12	expcom 116	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}))
14		breq1 4086	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑦 → (𝑒 ≈ 2o ↔ 𝑦 ≈ 2o))
15	14	rspcv 2903	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o → 𝑦 ≈ 2o))
16		breq1 4086	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 1o ↔ 𝑦 ≈ 1o))
17		breq1 4086	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 2o ↔ 𝑦 ≈ 2o))
18	16, 17	orbi12d 798	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 1o ∨ 𝑥 ≈ 2o) ↔ (𝑦 ≈ 1o ∨ 𝑦 ≈ 2o)))
19	18	elrab 2959	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} ↔ (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (𝑦 ≈ 1o ∨ 𝑦 ≈ 2o)))
20	17	elrab 2959	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o} ↔ (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑦 ≈ 2o))
21	20	simplbi2 385	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝒫 (Vtx‘𝐺) → (𝑦 ≈ 2o → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o}))
22	21	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (𝑦 ≈ 1o ∨ 𝑦 ≈ 2o)) → (𝑦 ≈ 2o → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o}))
23	19, 22	sylbi 121	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} → (𝑦 ≈ 2o → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o}))
24	15, 23	syl9 72	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} → (∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o})))
25	13, 24	syld 45	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} → (∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o})))
26	25	com13 80	. . . . . . . . . . . 12 ⊢ (∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} → (𝑦 ∈ ran (iEdg‘𝐺) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o})))
27	26	imp 124	. . . . . . . . . . 11 ⊢ ((∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o ∧ ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}) → (𝑦 ∈ ran (iEdg‘𝐺) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o}))
28	27	ssrdv 3230	. . . . . . . . . 10 ⊢ ((∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o ∧ ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)}) → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o})
29	28	ex 115	. . . . . . . . 9 ⊢ (∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o}))
30	11, 29	mpan9 281	. . . . . . . 8 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} ∧ ∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o) → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o})
31		f1ssr 5538	. . . . . . . 8 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} ∧ ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o}) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o})
32	30, 31	syldan 282	. . . . . . 7 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} ∧ ∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o})
33	32	ex 115	. . . . . 6 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} → (∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o}))
34	10, 33	syl 14	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o}))
35	7, 34	sylbid 150	. . . 4 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o}))
36	35	imp 124	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o})
37	8, 9	isusgren 15956	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o}))
38	37	adantr 276	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ 𝑥 ≈ 2o}))
39	36, 38	mpbird 167	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o) → 𝐺 ∈ USGraph)
40	5, 39	impbii 126	1 ⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   ∈ wcel 2200  ∀wral 2508  {crab 2512   ⊆ wss 3197  𝒫 cpw 3649   class class class wbr 4083  dom cdm 4719  ran crn 4720  –1-1→wf1 5315  ‘cfv 5318  1_oc1o 6555  2_oc2o 6556   ≈ cen 6885  Vtxcvtx 15813  iEdgciedg 15814  Edgcedg 15858  USPGraphcuspgr 15951  USGraphcusgr 15952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-sub 8319  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-dec 9579  df-ndx 13035  df-slot 13036  df-base 13038  df-edgf 15806  df-vtx 15815  df-iedg 15816  df-edg 15859  df-uspgren 15953  df-usgren 15954

This theorem is referenced by: (None)