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Metamath Proof Explorer

Theorem usgruspgr

Description: A simple graph is a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017) (Revised by AV, 15-Oct-2020)

Ref Expression
Assertion usgruspgr ( 𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2 eqid ⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
3 1 2 isusgr ⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
4 2re ⊢ 2 ∈ ℝ
5 4 eqlei2 ⊢ ( ( ♯ ‘ 𝑥 ) = 2 → ( ♯ ‘ 𝑥 ) ≤ 2 )
6 5 a1i ⊢ ( 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) → ( ( ♯ ‘ 𝑥 ) = 2 → ( ♯ ‘ 𝑥 ) ≤ 2 ) )
7 6 ss2rabi ⊢ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ⊆ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 }
8 f1ss ⊢ ( ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ∧ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ⊆ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
9 7 8 mpan2 ⊢ ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
10 3 9 biimtrdi ⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
11 1 2 isuspgr ⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ USPGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
12 10 11 sylibrd ⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ USGraph → 𝐺 ∈ USPGraph ) )
13 12 pm2.43i ⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )