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Theorem edgiedgb 27891
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 27886 . . . 4 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 edgiedgb.i . . . . . 6 𝐼 = (iEdg‘𝐺)
32eqcomi 2745 . . . . 5 (iEdg‘𝐺) = 𝐼
43rneqi 5890 . . . 4 ran (iEdg‘𝐺) = ran 𝐼
51, 4eqtri 2764 . . 3 (Edg‘𝐺) = ran 𝐼
65eleq2i 2829 . 2 (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼)
7 elrnrexdmb 7036 . 2 (Fun 𝐼 → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
86, 7bitrid 282 1 (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  wrex 3071  dom cdm 5631  ran crn 5632  Fun wfun 6487  cfv 6493  iEdgciedg 27834  Edgcedg 27884
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6445  df-fun 6495  df-fn 6496  df-fv 6501  df-edg 27885
This theorem is referenced by:  uhgredgiedgb  27963
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