Theorem iedgval 27092

Description: The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)

Assertion

Ref	Expression
iedgval	⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))

Proof of Theorem iedgval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2825	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2		fveq2 6717	. . . 4 ⊢ (𝑔 = 𝐺 → (2nd ‘𝑔) = (2nd ‘𝐺))
3		fveq2 6717	. . . 4 ⊢ (𝑔 = 𝐺 → (.ef‘𝑔) = (.ef‘𝐺))
4	1, 2, 3	ifbieq12d 4467	. . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)))
5		df-iedg 27090	. . 3 ⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
6		fvex 6730	. . . 4 ⊢ (2nd ‘𝐺) ∈ V
7		fvex 6730	. . . 4 ⊢ (.ef‘𝐺) ∈ V
8	6, 7	ifex 4489	. . 3 ⊢ if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) ∈ V
9	4, 5, 8	fvmpt 6818	. 2 ⊢ (𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)))
10		fvprc 6709	. . 3 ⊢ (¬ 𝐺 ∈ V → (.ef‘𝐺) = ∅)
11		prcnel 3431	. . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
12	11	iffalsed 4450	. . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (.ef‘𝐺))
13		fvprc 6709	. . 3 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
14	10, 12, 13	3eqtr4rd 2788	. 2 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)))
15	9, 14	pm2.61i 185	1 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))

Syntax hints:  ¬ wn 3   = wceq 1543   ∈ wcel 2110  Vcvv 3408  ∅c0 4237  ifcif 4439   × cxp 5549  ‘cfv 6380  2^nd c2nd 7760  .efcedgf 27079  iEdgciedg 27088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-iedg 27090

This theorem is referenced by:  opiedgval  27097  funiedgdmge2val  27103  funiedgdm2val  27105  snstriedgval  27129  iedgval0  27131