 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iedgval Structured version   Visualization version   GIF version

Theorem iedgval 26106
 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), (2nd𝐺), (.ef‘𝐺))

Proof of Theorem iedgval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2836 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2 fveq2 6331 . . . 4 (𝑔 = 𝐺 → (2nd𝑔) = (2nd𝐺))
3 fveq2 6331 . . . 4 (𝑔 = 𝐺 → (.ef‘𝑔) = (.ef‘𝐺))
41, 2, 3ifbieq12d 4249 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (2nd𝑔), (.ef‘𝑔)) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))
5 df-iedg 26104 . . 3 iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd𝑔), (.ef‘𝑔)))
6 fvex 6341 . . . 4 (2nd𝐺) ∈ V
7 fvex 6341 . . . 4 (.ef‘𝐺) ∈ V
86, 7ifex 4292 . . 3 if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)) ∈ V
94, 5, 8fvmpt 6423 . 2 (𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))
10 fvprc 6325 . . 3 𝐺 ∈ V → (.ef‘𝐺) = ∅)
11 prcnel 3366 . . . 4 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
1211iffalsed 4233 . . 3 𝐺 ∈ V → if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)) = (.ef‘𝐺))
13 fvprc 6325 . . 3 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
1410, 12, 133eqtr4rd 2814 . 2 𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))
159, 14pm2.61i 176 1 (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1629   ∈ wcel 2143  Vcvv 3348  ∅c0 4060  ifcif 4222   × cxp 5246  ‘cfv 6030  2nd c2nd 7312  .efcedgf 26094  iEdgciedg 26102 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ral 3064  df-rex 3065  df-rab 3068  df-v 3350  df-sbc 3585  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-iota 5993  df-fun 6032  df-fv 6038  df-iedg 26104 This theorem is referenced by:  opiedgval  26113  funiedgdmge2val  26119  funiedgdm2val  26121  snstriedgval  26157  iedgval0  26159
 Copyright terms: Public domain W3C validator