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Theorem edgiedgb 28047
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 28042 . . . 4 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 edgiedgb.i . . . . . 6 𝐼 = (iEdg‘𝐺)
32eqcomi 2746 . . . . 5 (iEdg‘𝐺) = 𝐼
43rneqi 5897 . . . 4 ran (iEdg‘𝐺) = ran 𝐼
51, 4eqtri 2765 . . 3 (Edg‘𝐺) = ran 𝐼
65eleq2i 2830 . 2 (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼)
7 elrnrexdmb 7045 . 2 (Fun 𝐼 → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
86, 7bitrid 283 1 (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wrex 3074  dom cdm 5638  ran crn 5639  Fun wfun 6495  cfv 6501  iEdgciedg 27990  Edgcedg 28040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509  df-edg 28041
This theorem is referenced by:  uhgredgiedgb  28119
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