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 )