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Theorem isfrgra 26517
 Description: The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
isfrgra ((𝑉𝑋𝐸𝑌) → (𝑉 FriendGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
Distinct variable groups:   𝑘,𝑉,𝑙,𝑥   𝑘,𝐸,𝑙,𝑥
Allowed substitution hints:   𝑋(𝑥,𝑘,𝑙)   𝑌(𝑥,𝑘,𝑙)

Proof of Theorem isfrgra
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4586 . . 3 (𝑣 = 𝑉 → (𝑣 USGrph 𝑒𝑉 USGrph 𝑒))
2 difeq1 3683 . . . . 5 (𝑣 = 𝑉 → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
3 reueq1 3117 . . . . 5 (𝑣 = 𝑉 → (∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒))
42, 3raleqbidv 3129 . . . 4 (𝑣 = 𝑉 → (∀𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒 ↔ ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒))
54raleqbi1dv 3123 . . 3 (𝑣 = 𝑉 → (∀𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒 ↔ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒))
61, 5anbi12d 743 . 2 (𝑣 = 𝑉 → ((𝑣 USGrph 𝑒 ∧ ∀𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒) ↔ (𝑉 USGrph 𝑒 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒)))
7 breq2 4587 . . 3 (𝑒 = 𝐸 → (𝑉 USGrph 𝑒𝑉 USGrph 𝐸))
8 rneq 5272 . . . . . 6 (𝑒 = 𝐸 → ran 𝑒 = ran 𝐸)
98sseq2d 3596 . . . . 5 (𝑒 = 𝐸 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒 ↔ {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
109reubidv 3103 . . . 4 (𝑒 = 𝐸 → (∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒 ↔ ∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
11102ralbidv 2972 . . 3 (𝑒 = 𝐸 → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒 ↔ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))
127, 11anbi12d 743 . 2 (𝑒 = 𝐸 → ((𝑉 USGrph 𝑒 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒) ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
13 df-frgra 26516 . 2 FriendGrph = {⟨𝑣, 𝑒⟩ ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝑒)}
146, 12, 13brabg 4919 1 ((𝑉𝑋𝐸𝑌) → (𝑉 FriendGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃!wreu 2898   ∖ cdif 3537   ⊆ wss 3540  {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-reu 2903  df-rab 2905  df-v 3175  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-cnv 5046  df-dm 5048  df-rn 5049  df-frgra 26516 This theorem is referenced by:  frisusgrapr  26518  frgra0v  26520  frgra1v  26525  frgra2v  26526  frgra3v  26529
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