Theorem iedgval 27274

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 2826	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2		fveq2 6756	. . . 4 ⊢ (𝑔 = 𝐺 → (2nd ‘𝑔) = (2nd ‘𝐺))
3		fveq2 6756	. . . 4 ⊢ (𝑔 = 𝐺 → (.ef‘𝑔) = (.ef‘𝐺))
4	1, 2, 3	ifbieq12d 4484	. . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)))
5		df-iedg 27272	. . 3 ⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
6		fvex 6769	. . . 4 ⊢ (2nd ‘𝐺) ∈ V
7		fvex 6769	. . . 4 ⊢ (.ef‘𝐺) ∈ V
8	6, 7	ifex 4506	. . 3 ⊢ if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) ∈ V
9	4, 5, 8	fvmpt 6857	. 2 ⊢ (𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)))
10		fvprc 6748	. . 3 ⊢ (¬ 𝐺 ∈ V → (.ef‘𝐺) = ∅)
11		prcnel 3445	. . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
12	11	iffalsed 4467	. . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (.ef‘𝐺))
13		fvprc 6748	. . 3 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
14	10, 12, 13	3eqtr4rd 2789	. 2 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)))
15	9, 14	pm2.61i 182	1 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  ifcif 4456   × cxp 5578  ‘cfv 6418  2^nd c2nd 7803  .efcedgf 27259  iEdgciedg 27270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-iedg 27272

This theorem is referenced by:  opiedgval  27279  funiedgdmge2val  27285  funiedgdm2val  27287  snstriedgval  27311  iedgval0  27313