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Metamath Proof Explorer

Theorem iscplgr

Description: The property of being a complete graph. (Contributed by AV, 1-Nov-2020)

Ref Expression
Hypothesis cplgruvtxb.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion iscplgr ( 𝐺 ∈ 𝑊 → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 cplgruvtxb.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 1 cplgruvtxb ⊢ ( 𝐺 ∈ 𝑊 → ( 𝐺 ∈ ComplGraph ↔ ( UnivVtx ‘ 𝐺 ) = 𝑉 ) )
3 eqss ⊢ ( ( UnivVtx ‘ 𝐺 ) = 𝑉 ↔ ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉 ∧ 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) )
4 1 uvtxssvtx ⊢ ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉
5 dfss3 ⊢ ( 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) )
6 5 anbi2i ⊢ ( ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉 ∧ 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) ↔ ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉 ∧ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )
7 4 6 mpbiran ⊢ ( ( ( UnivVtx ‘ 𝐺 ) ⊆ 𝑉 ∧ 𝑉 ⊆ ( UnivVtx ‘ 𝐺 ) ) ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) )
8 3 7 bitri ⊢ ( ( UnivVtx ‘ 𝐺 ) = 𝑉 ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) )
9 2 8 bitrdi ⊢ ( 𝐺 ∈ 𝑊 → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) )