Theorem iedgval 29636

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 2852	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2		fveq2 6916	. . . 4 ⊢ (𝑔 = 𝐺 → (2nd ‘𝑔) = (2nd ‘𝐺))
3		fveq2 6916	. . . 4 ⊢ (𝑔 = 𝐺 → (.ef‘𝑔) = (.ef‘𝐺))
4	1, 2, 3	ifbieq12d 4514	. . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)))
5		df-iedg 29634	. . 3 ⊢ iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd ‘𝑔), (.ef‘𝑔)))
6		fvex 6929	. . . 4 ⊢ (2nd ‘𝐺) ∈ V
7		fvex 6929	. . . 4 ⊢ (.ef‘𝐺) ∈ V
8	6, 7	ifex 4536	. . 3 ⊢ if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) ∈ V
9	4, 5, 8	fvmpt 7024	. 2 ⊢ (𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)))
10		fvprc 6908	. . 3 ⊢ (¬ 𝐺 ∈ V → (.ef‘𝐺) = ∅)
11		prcnel 3479	. . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
12	11	iffalsed 4496	. . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (.ef‘𝐺))
13		fvprc 6908	. . 3 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
14	10, 12, 13	3eqtr4rd 2809	. 2 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)))
15	9, 14	pm2.61i 184	1 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  ifcif 4485   × cxp 5656  ‘cfv 6566  2^nd c2nd 8034  .efcedgf 29623  iEdgciedg 29632

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-iedg 29634

This theorem is used by:  opiedgval  29641  funiedgdmge2val  29647  funiedgdm2val  29649  snstriedgval  29673  iedgval0  29675