Metamath Proof Explorer


Theorem subgruhgredgd

Description: An edge of a subgraph of a hypergraph is a nonempty subset of its vertices. (Contributed by AV, 17-Nov-2020) (Revised by AV, 21-Nov-2020)

Ref Expression
Hypotheses subgruhgredgd.v 𝑉 = ( Vtx ‘ 𝑆 )
subgruhgredgd.i 𝐼 = ( iEdg ‘ 𝑆 )
subgruhgredgd.g ( 𝜑𝐺 ∈ UHGraph )
subgruhgredgd.s ( 𝜑𝑆 SubGraph 𝐺 )
subgruhgredgd.x ( 𝜑𝑋 ∈ dom 𝐼 )
Assertion subgruhgredgd ( 𝜑 → ( 𝐼𝑋 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) )

Proof

Step Hyp Ref Expression
1 subgruhgredgd.v 𝑉 = ( Vtx ‘ 𝑆 )
2 subgruhgredgd.i 𝐼 = ( iEdg ‘ 𝑆 )
3 subgruhgredgd.g ( 𝜑𝐺 ∈ UHGraph )
4 subgruhgredgd.s ( 𝜑𝑆 SubGraph 𝐺 )
5 subgruhgredgd.x ( 𝜑𝑋 ∈ dom 𝐼 )
6 eqid ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
7 eqid ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
8 eqid ( Edg ‘ 𝑆 ) = ( Edg ‘ 𝑆 )
9 1 6 2 7 8 subgrprop2 ( 𝑆 SubGraph 𝐺 → ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) )
10 4 9 syl ( 𝜑 → ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) )
11 simpr3 ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 )
12 subgruhgrfun ( ( 𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺 ) → Fun ( iEdg ‘ 𝑆 ) )
13 3 4 12 syl2anc ( 𝜑 → Fun ( iEdg ‘ 𝑆 ) )
14 2 dmeqi dom 𝐼 = dom ( iEdg ‘ 𝑆 )
15 5 14 eleqtrdi ( 𝜑𝑋 ∈ dom ( iEdg ‘ 𝑆 ) )
16 13 15 jca ( 𝜑 → ( Fun ( iEdg ‘ 𝑆 ) ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) )
17 16 adantr ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( Fun ( iEdg ‘ 𝑆 ) ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) )
18 2 fveq1i ( 𝐼𝑋 ) = ( ( iEdg ‘ 𝑆 ) ‘ 𝑋 )
19 fvelrn ( ( Fun ( iEdg ‘ 𝑆 ) ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( ( iEdg ‘ 𝑆 ) ‘ 𝑋 ) ∈ ran ( iEdg ‘ 𝑆 ) )
20 18 19 eqeltrid ( ( Fun ( iEdg ‘ 𝑆 ) ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( 𝐼𝑋 ) ∈ ran ( iEdg ‘ 𝑆 ) )
21 17 20 syl ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼𝑋 ) ∈ ran ( iEdg ‘ 𝑆 ) )
22 edgval ( Edg ‘ 𝑆 ) = ran ( iEdg ‘ 𝑆 )
23 21 22 eleqtrrdi ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼𝑋 ) ∈ ( Edg ‘ 𝑆 ) )
24 11 23 sseldd ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼𝑋 ) ∈ 𝒫 𝑉 )
25 7 uhgrfun ( 𝐺 ∈ UHGraph → Fun ( iEdg ‘ 𝐺 ) )
26 3 25 syl ( 𝜑 → Fun ( iEdg ‘ 𝐺 ) )
27 26 adantr ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → Fun ( iEdg ‘ 𝐺 ) )
28 simpr2 ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → 𝐼 ⊆ ( iEdg ‘ 𝐺 ) )
29 5 adantr ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → 𝑋 ∈ dom 𝐼 )
30 funssfv ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) = ( 𝐼𝑋 ) )
31 30 eqcomd ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼𝑋 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) )
32 27 28 29 31 syl3anc ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼𝑋 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) )
33 3 adantr ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → 𝐺 ∈ UHGraph )
34 26 funfnd ( 𝜑 → ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) )
35 34 adantr ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) )
36 subgreldmiedg ( ( 𝑆 SubGraph 𝐺𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) )
37 4 15 36 syl2anc ( 𝜑𝑋 ∈ dom ( iEdg ‘ 𝐺 ) )
38 37 adantr ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) )
39 7 uhgrn0 ( ( 𝐺 ∈ UHGraph ∧ ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ≠ ∅ )
40 33 35 38 39 syl3anc ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ≠ ∅ )
41 32 40 eqnetrd ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼𝑋 ) ≠ ∅ )
42 eldifsn ( ( 𝐼𝑋 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ↔ ( ( 𝐼𝑋 ) ∈ 𝒫 𝑉 ∧ ( 𝐼𝑋 ) ≠ ∅ ) )
43 24 41 42 sylanbrc ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼𝑋 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) )
44 10 43 mpdan ( 𝜑 → ( 𝐼𝑋 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) )