Metamath Proof Explorer

Theorem isushgr

Description: The predicate "is an undirected simple hypergraph." (Contributed by AV, 19-Jan-2020) (Revised by AV, 9-Oct-2020)

Ref Expression
Hypotheses isuhgr.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
isuhgr.e ⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion isushgr ( 𝐺 ∈ 𝑈 → ( 𝐺 ∈ USHGraph ↔ 𝐸 : dom 𝐸 –1-1→ ( 𝒫 𝑉 ∖ { ∅ } ) ) )

Proof

Step Hyp Ref Expression
1 isuhgr.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 isuhgr.e ⊢ 𝐸 = ( iEdg ‘ 𝐺 )
3 df-ushgr ⊢ USHGraph = { 𝑔 ∣ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } ) }
4 3 eleq2i ⊢ ( 𝐺 ∈ USHGraph ↔ 𝐺 ∈ { 𝑔 ∣ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 –1-1→ ( 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 f1eq123d ⊢ ( ℎ = 𝐺 → ( ( iEdg ‘ ℎ ) : dom ( iEdg ‘ ℎ ) –1-1→ ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) ↔ 𝐸 : dom 𝐸 –1-1→ ( 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 f1eq123d ⊢ ( ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) ∧ 𝑒 = ( iEdg ‘ ℎ ) ) → ( 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } ) ↔ ( iEdg ‘ ℎ ) : dom ( iEdg ‘ ℎ ) –1-1→ ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) ) )
28 18 20 27 sbcied2 ⊢ ( ( 𝑔 = ℎ ∧ 𝑣 = ( Vtx ‘ ℎ ) ) → ( [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } ) ↔ ( iEdg ‘ ℎ ) : dom ( iEdg ‘ ℎ ) –1-1→ ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) ) )
29 16 17 28 sbcied2 ⊢ ( 𝑔 = ℎ → ( [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } ) ↔ ( iEdg ‘ ℎ ) : dom ( iEdg ‘ ℎ ) –1-1→ ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) ) )
30 29 cbvabv ⊢ { 𝑔 ∣ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } ) } = { ℎ ∣ ( iEdg ‘ ℎ ) : dom ( iEdg ‘ ℎ ) –1-1→ ( 𝒫 ( Vtx ‘ ℎ ) ∖ { ∅ } ) }
31 15 30 elab2g ⊢ ( 𝐺 ∈ 𝑈 → ( 𝐺 ∈ { 𝑔 ∣ [ ( Vtx ‘ 𝑔 ) / 𝑣 ] [ ( iEdg ‘ 𝑔 ) / 𝑒 ] 𝑒 : dom 𝑒 –1-1→ ( 𝒫 𝑣 ∖ { ∅ } ) } ↔ 𝐸 : dom 𝐸 –1-1→ ( 𝒫 𝑉 ∖ { ∅ } ) ) )
32 4 31 syl5bb ⊢ ( 𝐺 ∈ 𝑈 → ( 𝐺 ∈ USHGraph ↔ 𝐸 : dom 𝐸 –1-1→ ( 𝒫 𝑉 ∖ { ∅ } ) ) )