Theorem isuhgr

Description: The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017) (Revised by AV, 9-Oct-2020)

	Ref	Expression
Hypotheses	isuhgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	isuhgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion	isuhgr	⊢ ( 𝐺 ∈ 𝑈 → ( 𝐺 ∈ UHGraph ↔ 𝐸 : dom 𝐸 ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ) )

Proof

Step	Hyp	Ref	Expression
1		isuhgr.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		isuhgr.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
3		df-uhgr	⊢ UHGraph = { 𝑔 ∣ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 ⟶ ( 𝒫 𝑣 ∖ { ∅ } ) }
4	3	eleq2i	⊢ ( 𝐺 ∈ UHGraph ↔ 𝐺 ∈ { 𝑔 ∣ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 ⟶ ( 𝒫 𝑣 ∖ { ∅ } ) } )
5		fveq2	⊢ ( ℎ = 𝐺 → ( iEdg ‘ ℎ ) = ( iEdg ‘ 𝐺 ) )
6	5 2	eqtr4di	⊢ ( ℎ = 𝐺 → ( iEdg ‘ ℎ ) = 𝐸 )
7	5	dmeqd	⊢ ( ℎ = 𝐺 → dom ( iEdg ‘ ℎ ) = dom ( iEdg ‘ 𝐺 ) )
8	2	eqcomi	⊢ ( iEdg ‘ 𝐺 ) = 𝐸
9	8	dmeqi	⊢ dom ( iEdg ‘ 𝐺 ) = dom 𝐸
10	7 9	eqtrdi	⊢ ( ℎ = 𝐺 → dom ( iEdg ‘ ℎ ) = dom 𝐸 )
11		fveq2	⊢ ( ℎ = 𝐺 → ( Vtx ‘ ℎ ) = ( Vtx ‘ 𝐺 ) )
12	11 1	eqtr4di	⊢ ( ℎ = 𝐺 → ( Vtx ‘ ℎ ) = 𝑉 )
13	12	pweqd	⊢ ( ℎ = 𝐺 → 𝒫 ( Vtx ‘ ℎ ) = 𝒫 𝑉 )
14	13	difeq1d	⊢ ( ℎ = 𝐺 → ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) = ( 𝒫 𝑉 ∖ { ∅ } ) )
15	6 10 14	feq123d	⊢ ( ℎ = 𝐺 → ( ( iEdg ‘ ℎ ) : dom ( iEdg ‘ ℎ ) ⟶ ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) ↔ 𝐸 : dom 𝐸 ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ) )
16		fvexd	⊢ ( 𝑔 = ℎ → ( Vtx ‘ 𝑔 ) ∈ V )
17		fveq2	⊢ ( 𝑔 = ℎ → ( Vtx ‘ 𝑔 ) = ( Vtx ‘ ℎ ) )
18		fvexd	⊢ ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) → ( iEdg ‘ 𝑔 ) ∈ V )
19		fveq2	⊢ ( 𝑔 = ℎ → ( iEdg ‘ 𝑔 ) = ( iEdg ‘ ℎ ) )
20	19	adantr	⊢ ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) → ( iEdg ‘ 𝑔 ) = ( iEdg ‘ ℎ ) )
21		simpr	⊢ ( ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) ∧ 𝑒 = ( iEdg ‘ ℎ ) ) → 𝑒 = ( iEdg ‘ ℎ ) )
22	21	dmeqd	⊢ ( ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) ∧ 𝑒 = ( iEdg ‘ ℎ ) ) → dom 𝑒 = dom ( iEdg ‘ ℎ ) )
23		simpr	⊢ ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) → 𝑣 = ( Vtx ‘ ℎ ) )
24	23	pweqd	⊢ ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) → 𝒫 𝑣 = 𝒫 ( Vtx ‘ ℎ ) )
25	24	difeq1d	⊢ ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) → ( 𝒫 𝑣 ∖ { ∅ } ) = ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) )
26	25	adantr	⊢ ( ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) ∧ 𝑒 = ( iEdg ‘ ℎ ) ) → ( 𝒫 𝑣 ∖ { ∅ } ) = ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) )
27	21 22 26	feq123d	⊢ ( ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) ∧ 𝑒 = ( iEdg ‘ ℎ ) ) → ( 𝑒 : dom 𝑒 ⟶ ( 𝒫 𝑣 ∖ { ∅ } ) ↔ ( iEdg ‘ ℎ ) : dom ( iEdg ‘ ℎ ) ⟶ ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) ) )
28	18 20 27	sbcied2	⊢ ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) → ( [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 ⟶ ( 𝒫 𝑣 ∖ { ∅ } ) ↔ ( iEdg ‘ ℎ ) : dom ( iEdg ‘ ℎ ) ⟶ ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) ) )
29	16 17 28	sbcied2	⊢ ( 𝑔 = ℎ → ( [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 ⟶ ( 𝒫 𝑣 ∖ { ∅ } ) ↔ ( iEdg ‘ ℎ ) : dom ( iEdg ‘ ℎ ) ⟶ ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) ) )
30	29	cbvabv	⊢ { 𝑔 ∣ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 ⟶ ( 𝒫 𝑣 ∖ { ∅ } ) } = { ℎ ∣ ( iEdg ‘ ℎ ) : dom ( iEdg ‘ ℎ ) ⟶ ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) }
31	15 30	elab2g	⊢ ( 𝐺 ∈ 𝑈 → ( 𝐺 ∈ { 𝑔 ∣ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 ⟶ ( 𝒫 𝑣 ∖ { ∅ } ) } ↔ 𝐸 : dom 𝐸 ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ) )
32	4 31	syl5bb	⊢ ( 𝐺 ∈ 𝑈 → ( 𝐺 ∈ UHGraph ↔ 𝐸 : dom 𝐸 ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ) )