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Metamath Proof Explorer

Theorem subumgredg2

Description: An edge of a subgraph of a multigraph connects exactly two different vertices. (Contributed by AV, 26-Nov-2020)

Ref Expression
Hypotheses subumgredg2.v ⊢ 𝑉 = ( Vtx ‘ 𝑆 )
subumgredg2.i ⊢ 𝐼 = ( iEdg ‘ 𝑆 )
Assertion subumgredg2 ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ { 𝑒 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑒 ) = 2 } )

Proof

Step Hyp Ref Expression
1 subumgredg2.v ⊢ 𝑉 = ( Vtx ‘ 𝑆 )
2 subumgredg2.i ⊢ 𝐼 = ( iEdg ‘ 𝑆 )
3 fveqeq2 ⊢ ( 𝑒 = ( 𝐼 ‘ 𝑋 ) → ( ( ♯ ‘ 𝑒 ) = 2 ↔ ( ♯ ‘ ( 𝐼 ‘ 𝑋 ) ) = 2 ) )
4 umgruhgr ⊢ ( 𝐺 ∈ UMGraph → 𝐺 ∈ UHGraph )
5 4 3ad2ant2 ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝐺 ∈ UHGraph )
6 simp1 ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝑆 SubGraph 𝐺 )
7 simp3 ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ∈ dom 𝐼 )
8 1 2 5 6 7 subgruhgredgd ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) )
9 eqid ⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
10 9 uhgrfun ⊢ ( 𝐺 ∈ UHGraph → Fun ( iEdg ‘ 𝐺 ) )
11 4 10 syl ⊢ ( 𝐺 ∈ UMGraph → Fun ( iEdg ‘ 𝐺 ) )
12 11 3ad2ant2 ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → Fun ( iEdg ‘ 𝐺 ) )
13 eqid ⊢ ( Vtx ‘ 𝑆 ) = ( Vtx ‘ 𝑆 )
14 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
15 eqid ⊢ ( Edg ‘ 𝑆 ) = ( Edg ‘ 𝑆 )
16 13 14 2 9 15 subgrprop2 ⊢ ( 𝑆 SubGraph 𝐺 → ( ( Vtx ‘ 𝑆 ) ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 ( Vtx ‘ 𝑆 ) ) )
17 16 simp2d ⊢ ( 𝑆 SubGraph 𝐺 → 𝐼 ⊆ ( iEdg ‘ 𝐺 ) )
18 17 3ad2ant1 ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝐼 ⊆ ( iEdg ‘ 𝐺 ) )
19 funssfv ⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) = ( 𝐼 ‘ 𝑋 ) )
20 19 eqcomd ⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) )
21 12 18 7 20 syl3anc ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) )
22 21 fveq2d ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( ♯ ‘ ( 𝐼 ‘ 𝑋 ) ) = ( ♯ ‘ ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ) )
23 simp2 ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝐺 ∈ UMGraph )
24 2 dmeqi ⊢ dom 𝐼 = dom ( iEdg ‘ 𝑆 )
25 24 eleq2i ⊢ ( 𝑋 ∈ dom 𝐼 ↔ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) )
26 subgreldmiedg ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) )
27 26 ex ⊢ ( 𝑆 SubGraph 𝐺 → ( 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) )
28 25 27 biimtrid ⊢ ( 𝑆 SubGraph 𝐺 → ( 𝑋 ∈ dom 𝐼 → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) )
29 28 a1d ⊢ ( 𝑆 SubGraph 𝐺 → ( 𝐺 ∈ UMGraph → ( 𝑋 ∈ dom 𝐼 → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) ) )
30 29 3imp ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) )
31 14 9 umgredg2 ⊢ ( ( 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ♯ ‘ ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ) = 2 )
32 23 30 31 syl2anc ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( ♯ ‘ ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ) = 2 )
33 22 32 eqtrd ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( ♯ ‘ ( 𝐼 ‘ 𝑋 ) ) = 2 )
34 3 8 33 elrabd ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ { 𝑒 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑒 ) = 2 } )
35 prprrab ⊢ { 𝑒 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑒 ) = 2 } = { 𝑒 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑒 ) = 2 }
36 34 35 eleqtrdi ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐺 ∈ UMGraph ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ { 𝑒 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑒 ) = 2 } )