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Theorem iedgval 26808
 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 2877 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2 fveq2 6650 . . . 4 (𝑔 = 𝐺 → (2nd𝑔) = (2nd𝐺))
3 fveq2 6650 . . . 4 (𝑔 = 𝐺 → (.ef‘𝑔) = (.ef‘𝐺))
41, 2, 3ifbieq12d 4452 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (2nd𝑔), (.ef‘𝑔)) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))
5 df-iedg 26806 . . 3 iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd𝑔), (.ef‘𝑔)))
6 fvex 6663 . . . 4 (2nd𝐺) ∈ V
7 fvex 6663 . . . 4 (.ef‘𝐺) ∈ V
86, 7ifex 4473 . . 3 if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)) ∈ V
94, 5, 8fvmpt 6750 . 2 (𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))
10 fvprc 6642 . . 3 𝐺 ∈ V → (.ef‘𝐺) = ∅)
11 prcnel 3465 . . . 4 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
1211iffalsed 4436 . . 3 𝐺 ∈ V → if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)) = (.ef‘𝐺))
13 fvprc 6642 . . 3 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
1410, 12, 133eqtr4rd 2844 . 2 𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))
159, 14pm2.61i 185 1 (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  ifcif 4425   × cxp 5518  ‘cfv 6327  2nd c2nd 7677  .efcedgf 26796  iEdgciedg 26804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6286  df-fun 6329  df-fv 6335  df-iedg 26806 This theorem is referenced by:  opiedgval  26813  funiedgdmge2val  26819  funiedgdm2val  26821  snstriedgval  26845  iedgval0  26847
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