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Theorem edgiedgb 28822
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 28817 . . . 4 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 edgiedgb.i . . . . . 6 𝐼 = (iEdg‘𝐺)
32eqcomi 2735 . . . . 5 (iEdg‘𝐺) = 𝐼
43rneqi 5930 . . . 4 ran (iEdg‘𝐺) = ran 𝐼
51, 4eqtri 2754 . . 3 (Edg‘𝐺) = ran 𝐼
65eleq2i 2819 . 2 (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼)
7 elrnrexdmb 7085 . 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 1533  wcel 2098  wrex 3064  dom cdm 5669  ran crn 5670  Fun wfun 6531  cfv 6537  iEdgciedg 28765  Edgcedg 28815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545  df-edg 28816
This theorem is referenced by:  uhgredgiedgb  28894
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