Theorem wrdupgr

Description: The property of being an undirected pseudograph, expressing the edges as "words". (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	wrdupgr	⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → ( 𝐺 ∈ UPGraph ↔ 𝐸 ∈ Word { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )

Proof

Step	Hyp	Ref	Expression
1		isupgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isupgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
3	1 2	isupgr	⊢ ( 𝐺 ∈ 𝑈 → ( 𝐺 ∈ UPGraph ↔ 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
4	3	adantr	⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → ( 𝐺 ∈ UPGraph ↔ 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
5		wrdf	⊢ ( 𝐸 ∈ Word 𝑋 → 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ 𝑋 )
6	5	adantl	⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ 𝑋 )
7	6	fdmd	⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → dom 𝐸 = ( 0 ..^ ( ♯ ‘ 𝐸 ) ) )
8	7	feq2d	⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → ( 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ↔ 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
9		iswrdi	⊢ ( 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } → 𝐸 ∈ Word { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
10		wrdf	⊢ ( 𝐸 ∈ Word { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } → 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
11	9 10	impbii	⊢ ( 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ↔ 𝐸 ∈ Word { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
12	8 11	bitrdi	⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → ( 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ↔ 𝐸 ∈ Word { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )
13	4 12	bitrd	⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → ( 𝐺 ∈ UPGraph ↔ 𝐸 ∈ Word { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) )