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Metamath Proof Explorer

Theorem subgreldmiedg

Description: An element of the domain of the edge function of a subgraph is an element of the domain of the edge function of the supergraph. (Contributed by AV, 20-Nov-2020)

Ref Expression
Assertion subgreldmiedg ( ( 𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( Vtx ‘ 𝑆 ) = ( Vtx ‘ 𝑆 )
2 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
3 eqid ⊢ ( iEdg ‘ 𝑆 ) = ( iEdg ‘ 𝑆 )
4 eqid ⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
5 eqid ⊢ ( Edg ‘ 𝑆 ) = ( Edg ‘ 𝑆 )
6 1 2 3 4 5 subgrprop2 ⊢ ( 𝑆 SubGraph 𝐺 → ( ( Vtx ‘ 𝑆 ) ⊆ ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑆 ) ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 ( Vtx ‘ 𝑆 ) ) )
7 dmss ⊢ ( ( iEdg ‘ 𝑆 ) ⊆ ( iEdg ‘ 𝐺 ) → dom ( iEdg ‘ 𝑆 ) ⊆ dom ( iEdg ‘ 𝐺 ) )
8 7 3ad2ant2 ⊢ ( ( ( Vtx ‘ 𝑆 ) ⊆ ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑆 ) ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 ( Vtx ‘ 𝑆 ) ) → dom ( iEdg ‘ 𝑆 ) ⊆ dom ( iEdg ‘ 𝐺 ) )
9 8 sseld ⊢ ( ( ( Vtx ‘ 𝑆 ) ⊆ ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑆 ) ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 ( Vtx ‘ 𝑆 ) ) → ( 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) )
10 6 9 syl ⊢ ( 𝑆 SubGraph 𝐺 → ( 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) )
11 10 imp ⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) )