MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgra1v Structured version   Visualization version   GIF version

Theorem frgra1v 26264
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 25606 . . 3 ({𝑉} USGrph 𝐸 → ({𝑉} ∈ V ∧ 𝐸 ∈ V))
2 simplr 787 . . . . 5 (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉𝑋) → {𝑉} USGrph 𝐸)
3 ral0 3931 . . . . . 6 𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸
4 sneq 4038 . . . . . . . . . . 11 (𝑘 = 𝑉 → {𝑘} = {𝑉})
54difeq2d 3594 . . . . . . . . . 10 (𝑘 = 𝑉 → ({𝑉} ∖ {𝑘}) = ({𝑉} ∖ {𝑉}))
6 difid 3805 . . . . . . . . . 10 ({𝑉} ∖ {𝑉}) = ∅
75, 6syl6eq 2564 . . . . . . . . 9 (𝑘 = 𝑉 → ({𝑉} ∖ {𝑘}) = ∅)
8 preq2 4116 . . . . . . . . . . . 12 (𝑘 = 𝑉 → {𝑥, 𝑘} = {𝑥, 𝑉})
98preq1d 4121 . . . . . . . . . . 11 (𝑘 = 𝑉 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝑉}, {𝑥, 𝑙}})
109sseq1d 3499 . . . . . . . . . 10 (𝑘 = 𝑉 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
1110reubidv 3007 . . . . . . . . 9 (𝑘 = 𝑉 → (∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
127, 11raleqbidv 3033 . . . . . . . 8 (𝑘 = 𝑉 → (∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸))
1312ralsng 4068 . . . . . . 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 480 . . . . . 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 246 . . . . 5 (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉𝑋) → ∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)
16 isfrgra 26256 . . . . . 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 757 . . . . 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 958 . . . 4 (((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) ∧ 𝑉𝑋) → {𝑉} FriendGrph 𝐸)
1918ex 448 . . 3 ((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) → (𝑉𝑋 → {𝑉} FriendGrph 𝐸))
201, 19mpancom 699 . 2 ({𝑉} USGrph 𝐸 → (𝑉𝑋 → {𝑉} FriendGrph 𝐸))
2120impcom 444 1 ((𝑉𝑋 ∧ {𝑉} USGrph 𝐸) → {𝑉} FriendGrph 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  wral 2800  ∃!wreu 2802  Vcvv 3077  cdif 3441  wss 3444  c0 3777  {csn 4028  {cpr 4030   class class class wbr 4481  ran crn 4933   USGrph cusg 25598   FriendGrph cfrgra 26254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542  df-xp 4938  df-rel 4939  df-cnv 4940  df-dm 4942  df-rn 4943  df-usgra 25601  df-frgra 26255
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator