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Theorem edgiedgb 26853
Description: A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.) (Revised by AV, 8-Dec-2021.)
Hypothesis
Ref Expression
edgiedgb.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
edgiedgb (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
Distinct variable groups:   𝑥,𝐸   𝑥,𝐼
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem edgiedgb
StepHypRef Expression
1 edgval 26848 . . . 4 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 edgiedgb.i . . . . . 6 𝐼 = (iEdg‘𝐺)
32eqcomi 2833 . . . . 5 (iEdg‘𝐺) = 𝐼
43rneqi 5794 . . . 4 ran (iEdg‘𝐺) = ran 𝐼
51, 4eqtri 2847 . . 3 (Edg‘𝐺) = ran 𝐼
65eleq2i 2907 . 2 (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼)
7 elrnrexdmb 6847 . 2 (Fun 𝐼 → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
86, 7syl5bb 286 1 (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  wrex 3134  dom cdm 5542  ran crn 5543  Fun wfun 6337  cfv 6343  iEdgciedg 26796  Edgcedg 26846
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-edg 26847
This theorem is referenced by:  uhgredgiedgb  26925
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