Theorem usgruspgrb 28195

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 28192	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2		edgusgr 28174	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝑒) = 2))
3	2	simprd 496	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (♯‘𝑒) = 2)
4	3	ralrimiva 3139	. . 3 ⊢ (𝐺 ∈ USGraph → ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2)
5	1, 4	jca 512	. 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2))
6		edgval 28063	. . . . . . 7 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
7	6	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
8	7	raleqdv 3311	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2 ↔ ∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2))
9		eqid 2731	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10		eqid 2731	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
11	9, 10	uspgrf 28168	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
12		f1f 6743	. . . . . . . . . 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 6681	. . . . . . . . 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 3942	. . . . . . . . . . . . . . 15 ⊢ ((ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝑦 ∈ ran (iEdg‘𝐺)) → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
15	14	expcom 414	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
16		fveqeq2 6856	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑦 → ((♯‘𝑒) = 2 ↔ (♯‘𝑦) = 2))
17	16	rspcv 3578	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (♯‘𝑦) = 2))
18		fveq2 6847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
19	18	breq1d 5120	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝑦) ≤ 2))
20	19	elrab 3648	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) ≤ 2))
21		eldifi 4091	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝑦 ∈ 𝒫 (Vtx‘𝐺))
22	21	anim1i 615	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) = 2) → (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝑦) = 2))
23		fveqeq2 6856	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = 2 ↔ (♯‘𝑦) = 2))
24	23	elrab 3648	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2} ↔ (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝑦) = 2))
25	22, 24	sylibr 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) = 2) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2})
26	25	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
27	26	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) ≤ 2) → ((♯‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
28	20, 27	sylbi 216	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → ((♯‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
29	17, 28	syl9 77	. . . . . . . . . . . . . 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 47	. . . . . . . . . . . . 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 88	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 3953	. . . . . . . . . 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 413	. . . . . . . . 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 507	. . . . . . . 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 6750	. . . . . . . 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 591	. . . . . . 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 413	. . . . . 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 17	. . . . 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 239	. . . 4 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
41	40	imp 407	. . 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 28170	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
43	42	adantr 481	. . 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 256	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2) → 𝐺 ∈ USGraph)
45	5, 44	impbii 208	1 ⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  {crab 3405   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4287  𝒫 cpw 4565  {csn 4591   class class class wbr 5110  dom cdm 5638  ran crn 5639  –1-1→wf1 6498  ‘cfv 6501   ≤ cle 11199  2c2 12217  ♯chash 14240  Vtxcvtx 28010  iEdgciedg 28011  Edgcedg 28061  USPGraphcuspgr 28162  USGraphcusgr 28163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-n0 12423  df-z 12509  df-uz 12773  df-fz 13435  df-hash 14241  df-edg 28062  df-uspgr 28164  df-usgr 28165

This theorem is referenced by:  usgr1e  28256