Theorem usgruspgrb 26076

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 26073	. . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
2		edgusgr 26055	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝑒) = 2))
3	2	simprd 479	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (#‘𝑒) = 2)
4	3	ralrimiva 2966	. . 3 ⊢ (𝐺 ∈ USGraph → ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2)
5	1, 4	jca 554	. 2 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2))
6		edgval 25941	. . . . . . 7 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
7	6	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
8	7	raleqdv 3144	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2 ↔ ∀𝑒 ∈ ran (iEdg‘𝐺)(#‘𝑒) = 2))
9		eqid 2622	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10		eqid 2622	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
11	9, 10	uspgrf 26049	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
12		f1f 6101	. . . . . . . . . 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		frn 6053	. . . . . . . . . 10 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
14	12, 13	syl 17	. . . . . . . . 9 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
15		ssel2 3598	. . . . . . . . . . . . . . 15 ⊢ ((ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ∧ 𝑦 ∈ ran (iEdg‘𝐺)) → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
16	15	expcom 451	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
17		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝑦 → (#‘𝑒) = (#‘𝑦))
18	17	eqeq1d 2624	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑦 → ((#‘𝑒) = 2 ↔ (#‘𝑦) = 2))
19	18	rspcv 3305	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ran (iEdg‘𝐺) → (∀𝑒 ∈ ran (iEdg‘𝐺)(#‘𝑒) = 2 → (#‘𝑦) = 2))
20		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
21	20	breq1d 4663	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((#‘𝑥) ≤ 2 ↔ (#‘𝑦) ≤ 2))
22	21	elrab 3363	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) ≤ 2))
23		eldifi 3732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → 𝑦 ∈ 𝒫 (Vtx‘𝐺))
24	23	anim1i 592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) = 2) → (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝑦) = 2))
25	20	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → ((#‘𝑥) = 2 ↔ (#‘𝑦) = 2))
26	25	elrab 3363	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2} ↔ (𝑦 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝑦) = 2))
27	24, 26	sylibr 224	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) = 2) → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})
28	27	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((#‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
29	28	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (#‘𝑦) ≤ 2) → ((#‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
30	22, 29	sylbi 207	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ((#‘𝑦) = 2 → 𝑦 ∈ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
31	19, 30	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})))
32	16, 31	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})))
33	32	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})))
34	33	imp 445	. . . . . . . . . . 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}))
35	34	ssrdv 3609	. . . . . . . . . 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})
36	35	ex 450	. . . . . . . . 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}))
37	14, 36	mpan9 486	. . . . . . . 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})
38		f1ssr 6107	. . . . . . . 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})
39	37, 38	syldan 487	. . . . . . 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})
40	39	ex 450	. . . . . 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}))
41	11, 40	syl 17	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ ran (iEdg‘𝐺)(#‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
42	8, 41	sylbid 230	. . . 4 ⊢ (𝐺 ∈ USPGraph → (∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
43	42	imp 445	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})
44	9, 10	isusgrs 26051	. . . 4 ⊢ (𝐺 ∈ USPGraph → (𝐺 ∈ USGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2}))
45	44	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}))
46	43, 45	mpbird 247	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2) → 𝐺 ∈ USGraph )
47	5, 46	impbii 199	1 ⊢ (𝐺 ∈ USGraph ↔ (𝐺 ∈ USPGraph ∧ ∀𝑒 ∈ (Edg‘𝐺)(#‘𝑒) = 2))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177   class class class wbr 4653  dom cdm 5114  ran crn 5115  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888   ≤ cle 10075  2c2 11070  #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   USPGraph cuspgr 26043   USGraph cusgr 26044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-uspgr 26045  df-usgr 26046

This theorem is referenced by:  usgr1e  26137