Theorem usgruspgrben 16307

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 16304	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2		edgusgren 16284	. . . . 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 2617	. . 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 16180	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
7	6	raleqdv 2749	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o ↔ ∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o))
8		eqid 2234	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2234	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
10	8, 9	uspgrfen 16280	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
11		f1rn 5579	. . . . . . . . 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 3237	. . . . . . . . . . . . . . 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 4117	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑦 → (𝑒 ≈ 2o ↔ 𝑦 ≈ 2o))
15	14	rspcv 2919	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o → 𝑦 ≈ 2o))
16		breq1 4117	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 1o ↔ 𝑦 ≈ 1o))
17		breq1 4117	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 2o ↔ 𝑦 ≈ 2o))
18	16, 17	orbi12d 801	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 1o ∨ 𝑥 ≈ 2o) ↔ (𝑦 ≈ 1o ∨ 𝑦 ≈ 2o)))
19	18	elrab 2976	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} ↔ (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (𝑦 ≈ 1o ∨ 𝑦 ≈ 2o)))
20	17	elrab 2976	. . . . . . . . . . . . . . . . . 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 3248	. . . . . . . . . 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 5585	. . . . . . . 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 16279	. . . 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 716   ∈ wcel 2205  ∀wral 2522  {crab 2526   ⊆ wss 3214  𝒫 cpw 3674   class class class wbr 4114  dom cdm 4754  ran crn 4755  –1-1→wf1 5354  ‘cfv 5357  1_oc1o 6653  2_oc2o 6654   ≈ cen 6986  Vtxcvtx 16133  iEdgciedg 16134  Edgcedg 16178  USPGraphcuspgr 16274  USGraphcusgr 16275

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-uspgren 16276  df-usgren 16277

This theorem is referenced by:  usgr1e  16362