Metamath Proof Explorer


Theorem upgrle

Description: An edge of an undirected pseudograph has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015) (Revised by AV, 10-Oct-2020)

Ref Expression
Hypotheses isupgr.v 𝑉 = ( Vtx ‘ 𝐺 )
isupgr.e 𝐸 = ( iEdg ‘ 𝐺 )
Assertion upgrle ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴 ) → ( ♯ ‘ ( 𝐸𝐹 ) ) ≤ 2 )

Proof

Step Hyp Ref Expression
1 isupgr.v 𝑉 = ( Vtx ‘ 𝐺 )
2 isupgr.e 𝐸 = ( iEdg ‘ 𝐺 )
3 1 2 upgrfn ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ) → 𝐸 : 𝐴 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
4 3 ffvelrnda ( ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ) ∧ 𝐹𝐴 ) → ( 𝐸𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
5 4 3impa ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴 ) → ( 𝐸𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
6 fveq2 ( 𝑥 = ( 𝐸𝐹 ) → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ ( 𝐸𝐹 ) ) )
7 6 breq1d ( 𝑥 = ( 𝐸𝐹 ) → ( ( ♯ ‘ 𝑥 ) ≤ 2 ↔ ( ♯ ‘ ( 𝐸𝐹 ) ) ≤ 2 ) )
8 7 elrab ( ( 𝐸𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ↔ ( ( 𝐸𝐹 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∧ ( ♯ ‘ ( 𝐸𝐹 ) ) ≤ 2 ) )
9 8 simprbi ( ( 𝐸𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } → ( ♯ ‘ ( 𝐸𝐹 ) ) ≤ 2 )
10 5 9 syl ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴 ) → ( ♯ ‘ ( 𝐸𝐹 ) ) ≤ 2 )