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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 )