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	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (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.

Step	Hyp	Ref	Expression
1		breq12 4588	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 USGrph 𝑒 ↔ 𝑉 USGrph 𝐸))
2		simpl 472	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
3		difeq1 3683	. . . . . 6 ⊢ (𝑣 = 𝑉 → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
4	3	adantr 480	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 ∖ {𝑘}) = (𝑉 ∖ {𝑘}))
5		rneq 5272	. . . . . . 7 ⊢ (𝑒 = 𝐸 → ran 𝑒 = ran 𝐸)
6	5	adantl 481	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ran 𝑒 = ran 𝐸)
7	6	eleq2d 2673	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ({𝑛, 𝑘} ∈ ran 𝑒 ↔ {𝑛, 𝑘} ∈ ran 𝐸))
8	4, 7	raleqbidv 3129	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∀𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒 ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
9	2, 8	raleqbidv 3129	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∀𝑘 ∈ 𝑣 ∀𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒 ↔ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸))
10	1, 9	anbi12d 743	. 2 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((𝑣 USGrph 𝑒 ∧ ∀𝑘 ∈ 𝑣 ∀𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒) ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))
11		df-cusgra 25950	. 2 ⊢ ComplUSGrph = {⟨𝑣, 𝑒⟩ ∣ (𝑣 USGrph 𝑒 ∧ ∀𝑘 ∈ 𝑣 ∀𝑛 ∈ (𝑣 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝑒)}
12	10, 11	brabga 4914	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 USGrph 𝐸 ∧ ∀𝑘 ∈ 𝑉 ∀𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ ran 𝐸)))

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