Theorem usgruspgrb 28709

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 28706	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
2		edgusgr 28688	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝑒) = 2))
3	2	simprd 495	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (♯‘𝑒) = 2)
4	3	ralrimiva 3145	. . 3 ⊢ (𝐺 ∈ USGraph → ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2)
5	1, 4	jca 511	. 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(♯‘𝑒) = 2))
6		edgval 28577	. . . . . . 7 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
7	6	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
8	7	raleqdv 3324	. . . . 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 28682	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
12		f1f 6787	. . . . . . . . . 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 6725	. . . . . . . . 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 3977	. . . . . . . . . . . . . . 15 ⊢ ((ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ∧ 𝑦 ∈ ran (iEdg‘𝐺)) → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
15	14	expcom 413	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
16		fveqeq2 6900	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑦 → ((♯‘𝑒) = 2 ↔ (♯‘𝑦) = 2))
17	16	rspcv 3608	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (∀𝑒 ∈ ran (iEdg‘𝐺)(♯‘𝑒) = 2 → (♯‘𝑦) = 2))
18		fveq2 6891	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
19	18	breq1d 5158	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝑦) ≤ 2))
20	19	elrab 3683	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) ≤ 2))
21		eldifi 4126	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝑦 ∈ 𝒫 (Vtx‘𝐺))
22	21	anim1i 614	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝑦) = 2) → (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (♯‘𝑦) = 2))
23		fveqeq2 6900	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = 2 ↔ (♯‘𝑦) = 2))
24	23	elrab 3683	. . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((♯‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
27	26	adantr 480	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . 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 3988	. . . . . . . . . 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 412	. . . . . . . . 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 506	. . . . . . . 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 6794	. . . . . . . 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 590	. . . . . . 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 412	. . . . . 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 406	. . 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 28684	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}))
43	42	adantr 480	. . 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 257	. 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 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  {crab 3431   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  {csn 4628   class class class wbr 5148  dom cdm 5676  ran crn 5677  –1-1→wf1 6540  ‘cfv 6543   ≤ cle 11254  2c2 12272  ♯chash 14295  Vtxcvtx 28524  iEdgciedg 28525  Edgcedg 28575  USPGraphcuspgr 28676  USGraphcusgr 28677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-hash 14296  df-edg 28576  df-uspgr 28678  df-usgr 28679

This theorem is referenced by:  usgr1e  28770