Theorem frgra1v 26525

Description: Any graph with only one vertex is a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)

Assertion

Ref	Expression
frgra1v	⊢ ((𝑉 ∈ 𝑋 ∧ {𝑉} USGrph 𝐸) → {𝑉} FriendGrph 𝐸)

Proof of Theorem frgra1v

Dummy variables 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrav 25867	. . 3 ⊢ ({𝑉} USGrph 𝐸 → ({𝑉} ∈ V ∧ 𝐸 ∈ V))
2		simplr 788	. . . . 5 ⊢ (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉 ∈ 𝑋) → {𝑉} USGrph 𝐸)
3		ral0 4028	. . . . . 6 ⊢ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸
4		sneq 4135	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑉 → {𝑘} = {𝑉})
5	4	difeq2d 3690	. . . . . . . . . 10 ⊢ (𝑘 = 𝑉 → ({𝑉} ∖ {𝑘}) = ({𝑉} ∖ {𝑉}))
6		difid 3902	. . . . . . . . . 10 ⊢ ({𝑉} ∖ {𝑉}) = ∅
7	5, 6	syl6eq 2660	. . . . . . . . 9 ⊢ (𝑘 = 𝑉 → ({𝑉} ∖ {𝑘}) = ∅)
8		preq2 4213	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑉 → {𝑥, 𝑘} = {𝑥, 𝑉})
9	8	preq1d 4218	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑉 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝑉}, {𝑥, 𝑙}})
10	9	sseq1d 3595	. . . . . . . . . 10 ⊢ (𝑘 = 𝑉 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
11	10	reubidv 3103	. . . . . . . . 9 ⊢ (𝑘 = 𝑉 → (∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
12	7, 11	raleqbidv 3129	. . . . . . . 8 ⊢ (𝑘 = 𝑉 → (∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
13	12	ralsng 4165	. . . . . . 7 ⊢ (𝑉 ∈ 𝑋 → (∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
14	13	adantl 481	. . . . . 6 ⊢ (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉 ∈ 𝑋) → (∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
15	3, 14	mpbiri 247	. . . . 5 ⊢ (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉 ∈ 𝑋) → ∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)
16		isfrgra 26517	. . . . . 6 ⊢ (({𝑉} ∈ V ∧ 𝐸 ∈ V) → ({𝑉} FriendGrph 𝐸 ↔ ({𝑉} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
17	16	ad2antrr 758	. . . . 5 ⊢ (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉 ∈ 𝑋) → ({𝑉} FriendGrph 𝐸 ↔ ({𝑉} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
18	2, 15, 17	mpbir2and 959	. . . 4 ⊢ (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉 ∈ 𝑋) → {𝑉} FriendGrph 𝐸)
19	18	ex 449	. . 3 ⊢ ((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) → (𝑉 ∈ 𝑋 → {𝑉} FriendGrph 𝐸))
20	1, 19	mpancom 700	. 2 ⊢ ({𝑉} USGrph 𝐸 → (𝑉 ∈ 𝑋 → {𝑉} FriendGrph 𝐸))
21	20	impcom 445	1 ⊢ ((𝑉 ∈ 𝑋 ∧ {𝑉} USGrph 𝐸) → {𝑉} FriendGrph 𝐸)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃!wreu 2898  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   USGrph cusg 25859   FriendGrph cfrgra 26515

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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-usgra 25862  df-frgra 26516

This theorem is referenced by: (None)