Theorem usgruspgrb 29470

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

Assertion

Ref	Expression
usgruspgrb	⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2))

Distinct variable group:   𝑒,𝐺

Proof of Theorem usgruspgrb

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

Step	Hyp	Ref	Expression
1		usgruspgr 29467	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2		edgusgr 29447	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝑒) = 2))
3	2	simprd 500	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (♯‘𝑒) = 2)
4	3	ralrimiva 3163	. . 3 ⊢ (𝐺 ∈ USGraph → ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2)
5	1, 4	jca 520	. 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2))
6		edgval 29336	. . . . . . 7 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
7	6	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
8	7	raleqdv 3329	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2 ↔ ∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2))
9		eqid 2769	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10		eqid 2769	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
11	9, 10	uspgrf 29441	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
12		f1f 6772	. . . . . . . . . 10 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
13	12	frnd 6712	. . . . . . . . 9 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
14		ssel2 3940	. . . . . . . . . . . . . . 15 ⊢ ((ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝑦 ∈ ran (iEdg‘𝐺)) → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
15	14	expcom 418	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
16		fveqeq2 6888	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑦 → ((♯‘𝑒) = 2 ↔ (♯‘𝑦) = 2))
17	16	rspcv 3586	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (♯‘𝑦) = 2))
18		fveq2 6879	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
19	18	breq1d 5120	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝑦) ≤ 2))
20	19	elrab 3659	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) ≤ 2))
21		eldifi 4093	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝑦 ∈ 𝒫 (Vtx‘𝐺))
22	21	anim1i 626	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) = 2) → (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝑦) = 2))
23		fveqeq2 6888	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = 2 ↔ (♯‘𝑦) = 2))
24	23	elrab 3659	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2} ↔ (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝑦) = 2))
25	22, 24	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) = 2) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
26	25	ex 417	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
27	26	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) ≤ 2) → ((♯‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
28	20, 27	sylbi 220	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → ((♯‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
29	17, 28	syl9 78	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})))
30	15, 29	syld 48	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})))
31	30	com13 89	. . . . . . . . . . . 12 ⊢ (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (𝑦 ∈ ran (iEdg‘𝐺) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})))
32	31	imp 411	. . . . . . . . . . 11 ⊢ ((∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 ∧ ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) → (𝑦 ∈ ran (iEdg‘𝐺) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
33	32	ssrdv 3951	. . . . . . . . . 10 ⊢ ((∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 ∧ ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}) → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
34	33	ex 417	. . . . . . . . 9 ⊢ (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
35	13, 34	mpan9 515	. . . . . . . 8 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ ∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2) → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
36		f1ssr 6780	. . . . . . . 8 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ ran (iEdg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
37	35, 36	syldan 602	. . . . . . 7 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ ∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
38	37	ex 417	. . . . . 6 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
39	11, 38	syl 18	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
40	8, 39	sylbid 243	. . . 4 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
41	40	imp 411	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
42	9, 10	isusgrs 29443	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
43	42	adantr 485	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2) → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
44	41, 43	mpbird 260	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2) → 𝐺 ∈ USGraph)
45	5, 44	impbii 212	1 ⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4564  {csn 4591   class class class wbr 5110  dom cdm 5659  ran crn 5660  –1-1→wf1 6530  ‘cfv 6533   ≤ cle 11240  2c2 12291  ♯chash 14362  Vtxcvtx 29283  iEdgciedg 29284  Edgcedg 29334  USPGraphcuspgr 29435  USGraphcusgr 29436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-n0 12501  df-z 12588  df-uz 12859  df-fz 13532  df-hash 14363  df-edg 29335  df-uspgr 29437  df-usgr 29438

This theorem is referenced by:  usgr1e  29532