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Theorem iedgval0 39282
Description: Degenerated case 1 for edges: The set of indexed edges of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
Assertion
Ref Expression
iedgval0  |-  (iEdg `  (/) )  =  (/)

Proof of Theorem iedgval0
StepHypRef Expression
1 0nelxp 4884 . . 3  |-  -.  (/)  e.  ( _V  X.  _V )
21iffalsei 3903 . 2  |-  if (
(/)  e.  ( _V  X.  _V ) ,  ( 2nd `  (/) ) ,  (.ef `  (/) ) )  =  (.ef `  (/) )
3 0ex 4551 . . 3  |-  (/)  e.  _V
4 iedgval 39248 . . 3  |-  ( (/)  e.  _V  ->  (iEdg `  (/) )  =  if ( (/)  e.  ( _V  X.  _V ) ,  ( 2nd `  (/) ) ,  (.ef `  (/) ) ) )
53, 4ax-mp 5 . 2  |-  (iEdg `  (/) )  =  if (
(/)  e.  ( _V  X.  _V ) ,  ( 2nd `  (/) ) ,  (.ef `  (/) ) )
6 df-edgf 39238 . . 3  |- .ef  = Slot ; 1 8
76str0 15216 . 2  |-  (/)  =  (.ef
`  (/) )
82, 5, 73eqtr4i 2494 1  |-  (iEdg `  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455    e. wcel 1898   _Vcvv 3057   (/)c0 3743   ifcif 3893    X. cxp 4854   ` cfv 5605   2ndc2nd 6824   1c1 9571   8c8 10698  ;cdc 11085  .efcedgf 39237  iEdgciedg 39244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-iota 5569  df-fun 5607  df-fv 5613  df-slot 15180  df-edgf 39238  df-iedg 39246
This theorem is referenced by:  uhgr0  39309  usgr0  39464  0grsubgr  39496  0grrusgr  39741
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