Theorem usgruspgrben 15941

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 15938	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2		edgusgren 15918	. . . . 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 2581	. . 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 15817	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
7	6	raleqdv 2712	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)𝑒 ≈ 2o ↔ ∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o))
8		eqid 2207	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2207	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
10	8, 9	uspgrfen 15914	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
11		f1rn 5505	. . . . . . . . 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 3197	. . . . . . . . . . . . . . 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 4063	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑦 → (𝑒 ≈ 2o ↔ 𝑦 ≈ 2o))
15	14	rspcv 2881	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (∀𝑒 ∈ ran (iEdg‘𝐺)𝑒 ≈ 2o → 𝑦 ≈ 2o))
16		breq1 4063	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 1o ↔ 𝑦 ≈ 1o))
17		breq1 4063	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 2o ↔ 𝑦 ≈ 2o))
18	16, 17	orbi12d 795	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 1o ∨ 𝑥 ≈ 2o) ↔ (𝑦 ≈ 1o ∨ 𝑦 ≈ 2o)))
19	18	elrab 2937	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)} ↔ (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (𝑦 ≈ 1o ∨ 𝑦 ≈ 2o)))
20	17	elrab 2937	. . . . . . . . . . . . . . . . . 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 3208	. . . . . . . . . 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 5511	. . . . . . . 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 15913	. . . 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 710   ∈ wcel 2178  ∀wral 2486  {crab 2490   ⊆ wss 3175  𝒫 cpw 3627   class class class wbr 4060  dom cdm 4694  ran crn 4695  –1-1→wf1 5288  ‘cfv 5291  1_oc1o 6520  2_oc2o 6521   ≈ cen 6850  Vtxcvtx 15772  iEdgciedg 15773  Edgcedg 15815  USPGraphcuspgr 15908  USGraphcusgr 15909

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-cnex 8053  ax-resscn 8054  ax-1cn 8055  ax-1re 8056  ax-icn 8057  ax-addcl 8058  ax-addrcl 8059  ax-mulcl 8060  ax-addcom 8062  ax-mulcom 8063  ax-addass 8064  ax-mulass 8065  ax-distr 8066  ax-i2m1 8067  ax-1rid 8069  ax-0id 8070  ax-rnegex 8071  ax-cnre 8073

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-if 3581  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-br 4061  df-opab 4123  df-mpt 4124  df-id 4359  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-fv 5299  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-1st 6251  df-2nd 6252  df-sub 8282  df-inn 9074  df-2 9132  df-3 9133  df-4 9134  df-5 9135  df-6 9136  df-7 9137  df-8 9138  df-9 9139  df-n0 9333  df-dec 9542  df-ndx 12996  df-slot 12997  df-base 12999  df-edgf 15765  df-vtx 15774  df-iedg 15775  df-edg 15816  df-uspgren 15910  df-usgren 15911

This theorem is referenced by: (None)