Theorem usgruspgrben 16410

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 16407	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2		edgusgren 16387	. . . . 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 2623	. . 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 16283	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
7	6	raleqdv 2755	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o ↔ ∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o))
8		eqid 2238	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2238	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
10	8, 9	uspgrfen 16383	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
11		f1rn 5597	. . . . . . . . 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 3243	. . . . . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑦 → (𝑒 ≈ 2o ↔ 𝑦 ≈ 2o))
15	14	rspcv 2925	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o → 𝑦 ≈ 2o))
16		breq1 4131	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 1o ↔ 𝑦 ≈ 1o))
17		breq1 4131	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 2o ↔ 𝑦 ≈ 2o))
18	16, 17	orbi12d 805	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 1o ∨ 𝑥 ≈ 2o) ↔ (𝑦 ≈ 1o ∨ 𝑦 ≈ 2o)))
19	18	elrab 2982	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} ↔ (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (𝑦 ≈ 1o ∨ 𝑦 ≈ 2o)))
20	17	elrab 2982	. . . . . . . . . . . . . . . . . 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 3254	. . . . . . . . . 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 5603	. . . . . . . 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 16382	. . . 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 720   ∈ wcel 2209  ∀wral 2528  {crab 2532   ⊆ wss 3220  𝒫 cpw 3688   class class class wbr 4128  dom cdm 4772  ran crn 4773  –1-1→wf1 5372  ‘cfv 5375  1_oc1o 6674  2_oc2o 6675   ≈ cen 7014  Vtxcvtx 16236  iEdgciedg 16237  Edgcedg 16281  USPGraphcuspgr 16377  USGraphcusgr 16378

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-edg 16282  df-uspgren 16379  df-usgren 16380

This theorem is referenced by:  usgr1e  16465