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Theorem iscusgra 25985
 Description: The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
iscusgra ((𝑉𝑋𝐸𝑌) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
Distinct variable groups:   𝑘,𝑉,𝑛   𝑘,𝐸,𝑛
Allowed substitution hints:   𝑋(𝑘,𝑛)   𝑌(𝑘,𝑛)

Proof of Theorem iscusgra
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 4588 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 USGrph 𝑒𝑉 USGrph 𝐸))
2 simpl 472 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
3 difeq1 3683 . . . . . 6 (𝑣 = 𝑉 → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
43adantr 480 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
5 rneq 5272 . . . . . . 7 (𝑒 = 𝐸 → ran 𝑒 = ran 𝐸)
65adantl 481 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → ran 𝑒 = ran 𝐸)
76eleq2d 2673 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → ({𝑛, 𝑘} ∈ ran 𝑒 ↔ {𝑛, 𝑘} ∈ ran 𝐸))
84, 7raleqbidv 3129 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
92, 8raleqbidv 3129 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑘𝑣𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒 ↔ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
101, 9anbi12d 743 . 2 ((𝑣 = 𝑉𝑒 = 𝐸) → ((𝑣 USGrph 𝑒 ∧ ∀𝑘𝑣𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒) ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
11 df-cusgra 25950 . 2 ComplUSGrph = {⟨𝑣, 𝑒⟩ ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘𝑣𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒)}
1210, 11brabga 4914 1 ((𝑉𝑋𝐸𝑌) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039   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-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-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-cusgra 25950 This theorem is referenced by:  iscusgra0  25986  cusgra0v  25989  cusgra1v  25990  cusgra2v  25991  nbcusgra  25992  cusgra3v  25993  cusgraexi  25997  cusgrares  26001  cusgrauvtxb  26024  cusconngra  26204
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