Theorem iedgval0 26823

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

Step	Hyp	Ref	Expression
1		0nelxp 5582	. . 3 ⊢ ¬ ∅ ∈ (V × V)
2	1	iffalsei 4470	. 2 ⊢ if(∅ ∈ (V × V), (2nd ‘∅), (.ef‘∅)) = (.ef‘∅)
3		iedgval 26784	. 2 ⊢ (iEdg‘∅) = if(∅ ∈ (V × V), (2nd ‘∅), (.ef‘∅))
4		df-edgf 26773	. . 3 ⊢ .ef = Slot ;18
5	4	str0 16530	. 2 ⊢ ∅ = (.ef‘∅)
6	2, 3, 5	3eqtr4i 2853	1 ⊢ (iEdg‘∅) = ∅

Syntax hints:   = wceq 1536   ∈ wcel 2113  Vcvv 3491  ∅c0 4284  ifcif 4460   × cxp 5546  ‘cfv 6348  2^nd c2nd 7681  1c1 10531  8c8 11692  ;cdc 12092  .efcedgf 26772  iEdgciedg 26780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-sbc 3769  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-slot 16482  df-edgf 26773  df-iedg 26782

This theorem is referenced by:  uhgr0  26856  usgr0  27023  0grsubgr  27058  0grrusgr  27359