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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 𝐴 ∧ 𝐹 ∈ 𝐴 ) → ( ♯ ‘ ( 𝐸 ‘ 𝐹 ) ) ≤ 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 ffvelcdmda ⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ) ∧ 𝐹 ∈ 𝐴 ) → ( 𝐸 ‘ 𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
5 4 3impa ⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴 ) → ( 𝐸 ‘ 𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
6 fveq2 ⊢ ( 𝑥 = ( 𝐸 ‘ 𝐹 ) → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ ( 𝐸 ‘ 𝐹 ) ) )
7 6 breq1d ⊢ ( 𝑥 = ( 𝐸 ‘ 𝐹 ) → ( ( ♯ ‘ 𝑥 ) ≤ 2 ↔ ( ♯ ‘ ( 𝐸 ‘ 𝐹 ) ) ≤ 2 ) )
8 7 elrab ⊢ ( ( 𝐸 ‘ 𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ↔ ( ( 𝐸 ‘ 𝐹 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∧ ( ♯ ‘ ( 𝐸 ‘ 𝐹 ) ) ≤ 2 ) )
9 8 simprbi ⊢ ( ( 𝐸 ‘ 𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } → ( ♯ ‘ ( 𝐸 ‘ 𝐹 ) ) ≤ 2 )
10 5 9 syl ⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴 ) → ( ♯ ‘ ( 𝐸 ‘ 𝐹 ) ) ≤ 2 )