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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 ((𝑉𝑋 ∧ {𝑉} USGrph 𝐸) → {𝑉} FriendGrph 𝐸)

Proof of Theorem frgra1v
Dummy variables 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 25867 . . 3 ({𝑉} USGrph 𝐸 → ({𝑉} ∈ V ∧ 𝐸 ∈ V))
2 simplr 788 . . . . 5 (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉𝑋) → {𝑉} USGrph 𝐸)
3 ral0 4028 . . . . . 6 𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸
4 sneq 4135 . . . . . . . . . . 11 (𝑘 = 𝑉 → {𝑘} = {𝑉})
54difeq2d 3690 . . . . . . . . . 10 (𝑘 = 𝑉 → ({𝑉} ∖ {𝑘}) = ({𝑉} ∖ {𝑉}))
6 difid 3902 . . . . . . . . . 10 ({𝑉} ∖ {𝑉}) = ∅
75, 6syl6eq 2660 . . . . . . . . 9 (𝑘 = 𝑉 → ({𝑉} ∖ {𝑘}) = ∅)
8 preq2 4213 . . . . . . . . . . . 12 (𝑘 = 𝑉 → {𝑥, 𝑘} = {𝑥, 𝑉})
98preq1d 4218 . . . . . . . . . . 11 (𝑘 = 𝑉 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝑉}, {𝑥, 𝑙}})
109sseq1d 3595 . . . . . . . . . 10 (𝑘 = 𝑉 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
1110reubidv 3103 . . . . . . . . 9 (𝑘 = 𝑉 → (∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
127, 11raleqbidv 3129 . . . . . . . 8 (𝑘 = 𝑉 → (∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
1312ralsng 4165 . . . . . . 7 (𝑉𝑋 → (∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
1413adantl 481 . . . . . 6 (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉𝑋) → (∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
153, 14mpbiri 247 . . . . 5 (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉𝑋) → ∀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)))
1716ad2antrr 758 . . . . 5 (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉𝑋) → ({𝑉} FriendGrph 𝐸 ↔ ({𝑉} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
182, 15, 17mpbir2and 959 . . . 4 (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉𝑋) → {𝑉} FriendGrph 𝐸)
1918ex 449 . . 3 ((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) → (𝑉𝑋 → {𝑉} FriendGrph 𝐸))
201, 19mpancom 700 . 2 ({𝑉} USGrph 𝐸 → (𝑉𝑋 → {𝑉} FriendGrph 𝐸))
2120impcom 445 1 ((𝑉𝑋 ∧ {𝑉} USGrph 𝐸) → {𝑉} FriendGrph 𝐸)
 Colors of variables: wff setvar class 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)
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