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Theorem iedgval0 26392
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 5391 . . 3 ¬ ∅ ∈ (V × V)
21iffalsei 4317 . 2 if(∅ ∈ (V × V), (2nd ‘∅), (.ef‘∅)) = (.ef‘∅)
3 iedgval 26353 . 2 (iEdg‘∅) = if(∅ ∈ (V × V), (2nd ‘∅), (.ef‘∅))
4 df-edgf 26342 . . 3 .ef = Slot 18
54str0 16311 . 2 ∅ = (.ef‘∅)
62, 3, 53eqtr4i 2812 1 (iEdg‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1601  wcel 2107  Vcvv 3398  c0 4141  ifcif 4307   × cxp 5355  cfv 6137  2nd c2nd 7446  1c1 10275  8c8 11440  cdc 11849  .efcedgf 26341  iEdgciedg 26349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-slot 16263  df-edgf 26342  df-iedg 26351
This theorem is referenced by:  uhgr0  26425  usgr0  26594  0grsubgr  26629  0grrusgr  26931
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