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Theorem iscusgra 25724
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 4486 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 USGrph 𝑒𝑉 USGrph 𝐸))
2 simpl 471 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
3 difeq1 3587 . . . . . 6 (𝑣 = 𝑉 → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
43adantr 479 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
5 rneq 5163 . . . . . . 7 (𝑒 = 𝐸 → ran 𝑒 = ran 𝐸)
65adantl 480 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → ran 𝑒 = ran 𝐸)
76eleq2d 2577 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → ({𝑛, 𝑘} ∈ ran 𝑒 ↔ {𝑛, 𝑘} ∈ ran 𝐸))
84, 7raleqbidv 3033 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
92, 8raleqbidv 3033 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑘𝑣𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒 ↔ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
101, 9anbi12d 742 . 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 25689 . 2 ComplUSGrph = {⟨𝑣, 𝑒⟩ ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘𝑣𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒)}
1210, 11brabga 4808 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 194  wa 382   = wceq 1474  wcel 1938  wral 2800  cdif 3441  {csn 4028  {cpr 4030   class class class wbr 4481  ran crn 4933   USGrph cusg 25598   ComplUSGrph ccusgra 25686
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-rab 2809  df-v 3079  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-cnv 4940  df-dm 4942  df-rn 4943  df-cusgra 25689
This theorem is referenced by:  iscusgra0  25725  cusgra0v  25728  cusgra1v  25729  cusgra2v  25730  nbcusgra  25731  cusgra3v  25732  cusgraexi  25736  cusgrares  25740  cusgrauvtxb  25763  cusconngra  25943
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