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 syl6bi ( 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 )