Metamath Proof Explorer


Theorem edgfiedgval

Description: The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020) (Revised by AV, 12-Nov-2021)

Ref Expression
Hypotheses basvtxval.s
|- ( ph -> G Struct X )
basvtxval.d
|- ( ph -> 2 <_ ( # ` dom G ) )
edgfiedgval.e
|- ( ph -> E e. Y )
edgfiedgval.f
|- ( ph -> <. ( .ef ` ndx ) , E >. e. G )
Assertion edgfiedgval
|- ( ph -> ( iEdg ` G ) = E )

Proof

Step Hyp Ref Expression
1 basvtxval.s
 |-  ( ph -> G Struct X )
2 basvtxval.d
 |-  ( ph -> 2 <_ ( # ` dom G ) )
3 edgfiedgval.e
 |-  ( ph -> E e. Y )
4 edgfiedgval.f
 |-  ( ph -> <. ( .ef ` ndx ) , E >. e. G )
5 structn0fun
 |-  ( G Struct X -> Fun ( G \ { (/) } ) )
6 1 5 syl
 |-  ( ph -> Fun ( G \ { (/) } ) )
7 funiedgdmge2val
 |-  ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> ( iEdg ` G ) = ( .ef ` G ) )
8 6 2 7 syl2anc
 |-  ( ph -> ( iEdg ` G ) = ( .ef ` G ) )
9 edgfid
 |-  .ef = Slot ( .ef ` ndx )
10 structex
 |-  ( G Struct X -> G e. _V )
11 1 10 syl
 |-  ( ph -> G e. _V )
12 structfung
 |-  ( G Struct X -> Fun `' `' G )
13 1 12 syl
 |-  ( ph -> Fun `' `' G )
14 9 11 13 4 3 strfv2d
 |-  ( ph -> E = ( .ef ` G ) )
15 8 14 eqtr4d
 |-  ( ph -> ( iEdg ` G ) = E )