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Metamath Proof Explorer

Theorem subgrfun

Description: The edge function of a subgraph of a graph whose edge function is actually a function is a function. (Contributed by AV, 20-Nov-2020)

Ref Expression
Assertion subgrfun ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝑆 SubGraph 𝐺 ) → Fun ( iEdg ‘ 𝑆 ) )

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 funss ⊢ ( ( iEdg ‘ 𝑆 ) ⊆ ( iEdg ‘ 𝐺 ) → ( Fun ( iEdg ‘ 𝐺 ) → Fun ( iEdg ‘ 𝑆 ) ) )
8 7 3ad2ant2 ⊢ ( ( ( Vtx ‘ 𝑆 ) ⊆ ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑆 ) ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 ( Vtx ‘ 𝑆 ) ) → ( Fun ( iEdg ‘ 𝐺 ) → Fun ( iEdg ‘ 𝑆 ) ) )
9 6 8 syl ⊢ ( 𝑆 SubGraph 𝐺 → ( Fun ( iEdg ‘ 𝐺 ) → Fun ( iEdg ‘ 𝑆 ) ) )
10 9 impcom ⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝑆 SubGraph 𝐺 ) → Fun ( iEdg ‘ 𝑆 ) )