Theorem cusgrarn 25988

Description: In a complete simple graph, the range of the edge function consists of all the pairs with different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.)

Assertion

Ref	Expression
cusgrarn	⊢ (𝑉 ComplUSGrph 𝐸 → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})

Distinct variable groups:   𝑥,𝐸   𝑥,𝑉

Proof of Theorem cusgrarn

Dummy variables 𝑎 𝑏 𝑘 𝑙 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscusgra0 25986	. 2 ⊢ (𝑉 ComplUSGrph 𝐸 → (𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸))
2		usgraf0 25877	. . 3 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
3		f1f 6014	. . . . . . 7 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
4		df-f 5808	. . . . . . . 8 ⊢ (𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ (𝐸 Fn dom 𝐸 ∧ ran 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
5		ssel 3562	. . . . . . . . 9 ⊢ (ran 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (𝑒 ∈ ran 𝐸 → 𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
6	5	adantl 481	. . . . . . . 8 ⊢ ((𝐸 Fn dom 𝐸 ∧ ran 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) → (𝑒 ∈ ran 𝐸 → 𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
7	4, 6	sylbi 206	. . . . . . 7 ⊢ (𝐸:dom 𝐸⟶{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (𝑒 ∈ ran 𝐸 → 𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
8	3, 7	syl 17	. . . . . 6 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (𝑒 ∈ ran 𝐸 → 𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
9	8	adantr 480	. . . . 5 ⊢ ((𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸) → (𝑒 ∈ ran 𝐸 → 𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
10		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑥 = 𝑒 → (#‘𝑥) = (#‘𝑒))
11	10	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑥 = 𝑒 → ((#‘𝑥) = 2 ↔ (#‘𝑒) = 2))
12	11	elrab 3331	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ↔ (𝑒 ∈ 𝒫 𝑉 ∧ (#‘𝑒) = 2))
13		vex 3176	. . . . . . . . . . 11 ⊢ 𝑒 ∈ V
14		hash2prde 13109	. . . . . . . . . . 11 ⊢ ((𝑒 ∈ V ∧ (#‘𝑒) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑒 = {𝑎, 𝑏}))
15	13, 14	mpan 702	. . . . . . . . . 10 ⊢ ((#‘𝑒) = 2 → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑒 = {𝑎, 𝑏}))
16		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ∈ 𝒫 𝑉))
17		prex 4836	. . . . . . . . . . . . . . . . 17 ⊢ {𝑎, 𝑏} ∈ V
18	17	elpw 4114	. . . . . . . . . . . . . . . 16 ⊢ ({𝑎, 𝑏} ∈ 𝒫 𝑉 ↔ {𝑎, 𝑏} ⊆ 𝑉)
19		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑎 ∈ V
20		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑏 ∈ V
21	19, 20	prss 4291	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ↔ {𝑎, 𝑏} ⊆ 𝑉)
22	18, 21	bitr4i 266	. . . . . . . . . . . . . . 15 ⊢ ({𝑎, 𝑏} ∈ 𝒫 𝑉 ↔ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉))
23		simprl 790	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑒 = {𝑎, 𝑏} ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
24	23	anim1i 590	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑒 = {𝑎, 𝑏} ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑎 ≠ 𝑏) → (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑏))
25		eldifsn 4260	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (𝑉 ∖ {𝑏}) ↔ (𝑎 ∈ 𝑉 ∧ 𝑎 ≠ 𝑏))
26	24, 25	sylibr 223	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑒 = {𝑎, 𝑏} ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑎 ≠ 𝑏) → 𝑎 ∈ (𝑉 ∖ {𝑏}))
27		simplrr 797	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑒 = {𝑎, 𝑏} ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑎 ≠ 𝑏) → 𝑏 ∈ 𝑉)
28		sneq 4135	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑏 → {𝑘} = {𝑏})
29	28	difeq2d 3690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑏 → (𝑉 ∖ {𝑘}) = (𝑉 ∖ {𝑏}))
30		preq2 4213	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑏 → {𝑙, 𝑘} = {𝑙, 𝑏})
31	30	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑏 → ({𝑙, 𝑘} ∈ ran 𝐸 ↔ {𝑙, 𝑏} ∈ ran 𝐸))
32	29, 31	raleqbidv 3129	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑏 → (∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 ↔ ∀𝑙 ∈ (𝑉 ∖ {𝑏}){𝑙, 𝑏} ∈ ran 𝐸))
33	32	rspcv 3278	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ 𝑉 → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑏}){𝑙, 𝑏} ∈ ran 𝐸))
34	27, 33	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑒 = {𝑎, 𝑏} ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑎 ≠ 𝑏) → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → ∀𝑙 ∈ (𝑉 ∖ {𝑏}){𝑙, 𝑏} ∈ ran 𝐸))
35		preq1 4212	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑎 → {𝑙, 𝑏} = {𝑎, 𝑏})
36	35	eleq1d 2672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑎 → ({𝑙, 𝑏} ∈ ran 𝐸 ↔ {𝑎, 𝑏} ∈ ran 𝐸))
37	36	rspcv 3278	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ (𝑉 ∖ {𝑏}) → (∀𝑙 ∈ (𝑉 ∖ {𝑏}){𝑙, 𝑏} ∈ ran 𝐸 → {𝑎, 𝑏} ∈ ran 𝐸))
38	26, 34, 37	sylsyld 59	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑒 = {𝑎, 𝑏} ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑎 ≠ 𝑏) → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → {𝑎, 𝑏} ∈ ran 𝐸))
39		eleq1 2676	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑒 ∈ ran 𝐸 ↔ {𝑎, 𝑏} ∈ ran 𝐸))
40	39	bicomd 212	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 = {𝑎, 𝑏} → ({𝑎, 𝑏} ∈ ran 𝐸 ↔ 𝑒 ∈ ran 𝐸))
41	40	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑒 = {𝑎, 𝑏} ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({𝑎, 𝑏} ∈ ran 𝐸 ↔ 𝑒 ∈ ran 𝐸))
42	41	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑒 = {𝑎, 𝑏} ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑎 ≠ 𝑏) → ({𝑎, 𝑏} ∈ ran 𝐸 ↔ 𝑒 ∈ ran 𝐸))
43	38, 42	sylibd 228	. . . . . . . . . . . . . . . 16 ⊢ (((𝑒 = {𝑎, 𝑏} ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑎 ≠ 𝑏) → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → 𝑒 ∈ ran 𝐸))
44	43	exp31 628	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = {𝑎, 𝑏} → ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎 ≠ 𝑏 → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → 𝑒 ∈ ran 𝐸))))
45	22, 44	syl5bi 231	. . . . . . . . . . . . . 14 ⊢ (𝑒 = {𝑎, 𝑏} → ({𝑎, 𝑏} ∈ 𝒫 𝑉 → (𝑎 ≠ 𝑏 → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → 𝑒 ∈ ran 𝐸))))
46	16, 45	sylbid 229	. . . . . . . . . . . . 13 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑒 ∈ 𝒫 𝑉 → (𝑎 ≠ 𝑏 → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → 𝑒 ∈ ran 𝐸))))
47	46	com23 84	. . . . . . . . . . . 12 ⊢ (𝑒 = {𝑎, 𝑏} → (𝑎 ≠ 𝑏 → (𝑒 ∈ 𝒫 𝑉 → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → 𝑒 ∈ ran 𝐸))))
48	47	impcom 445	. . . . . . . . . . 11 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑒 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 𝑉 → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → 𝑒 ∈ ran 𝐸)))
49	48	exlimivv 1847	. . . . . . . . . 10 ⊢ (∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑒 = {𝑎, 𝑏}) → (𝑒 ∈ 𝒫 𝑉 → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → 𝑒 ∈ ran 𝐸)))
50	15, 49	syl 17	. . . . . . . . 9 ⊢ ((#‘𝑒) = 2 → (𝑒 ∈ 𝒫 𝑉 → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → 𝑒 ∈ ran 𝐸)))
51	50	impcom 445	. . . . . . . 8 ⊢ ((𝑒 ∈ 𝒫 𝑉 ∧ (#‘𝑒) = 2) → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → 𝑒 ∈ ran 𝐸))
52	12, 51	sylbi 206	. . . . . . 7 ⊢ (𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → 𝑒 ∈ ran 𝐸))
53	52	com12 32	. . . . . 6 ⊢ (∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸 → (𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝑒 ∈ ran 𝐸))
54	53	adantl 481	. . . . 5 ⊢ ((𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸) → (𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} → 𝑒 ∈ ran 𝐸))
55	9, 54	impbid 201	. . . 4 ⊢ ((𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸) → (𝑒 ∈ ran 𝐸 ↔ 𝑒 ∈ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
56	55	eqrdv 2608	. . 3 ⊢ ((𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸) → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
57	2, 56	sylan 487	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘}){𝑙, 𝑘} ∈ ran 𝐸) → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
58	1, 57	syl 17	1 ⊢ (𝑉 ComplUSGrph 𝐸 → ran 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  2c2 10947  #chash 12979   USGrph cusg 25859   ComplUSGrph ccusgra 25947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862  df-cusgra 25950

This theorem is referenced by:  cusgrafilem1  26007