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Theorem iedgval0 26153
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 5301 . . 3 ¬ ∅ ∈ (V × V)
21iffalsei 4241 . 2 if(∅ ∈ (V × V), (2nd ‘∅), (.ef‘∅)) = (.ef‘∅)
3 iedgval 26100 . 2 (iEdg‘∅) = if(∅ ∈ (V × V), (2nd ‘∅), (.ef‘∅))
4 df-edgf 26089 . . 3 .ef = Slot 18
54str0 16134 . 2 ∅ = (.ef‘∅)
62, 3, 53eqtr4i 2793 1 (iEdg‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wcel 2140  Vcvv 3341  c0 4059  ifcif 4231   × cxp 5265  cfv 6050  2nd c2nd 7334  1c1 10150  8c8 11289  cdc 11706  .efcedgf 26088  iEdgciedg 26096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-iota 6013  df-fun 6052  df-fv 6058  df-slot 16084  df-edgf 26089  df-iedg 26098
This theorem is referenced by:  uhgr0  26189  usgr0  26356  0grsubgr  26391  0grrusgr  26707
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