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Theorem edgiedgb 26168
 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 26162 . . . 4 (Edg‘𝐺) = ran (iEdg‘𝐺)
2 edgiedgb.i . . . . . 6 𝐼 = (iEdg‘𝐺)
32eqcomi 2780 . . . . 5 (iEdg‘𝐺) = 𝐼
43rneqi 5490 . . . 4 ran (iEdg‘𝐺) = ran 𝐼
51, 4eqtri 2793 . . 3 (Edg‘𝐺) = ran 𝐼
65eleq2i 2842 . 2 (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼)
7 elrnrexdmb 6507 . 2 (Fun 𝐼 → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
86, 7syl5bb 272 1 (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  dom cdm 5249  ran crn 5250  Fun wfun 6025  ‘cfv 6031  iEdgciedg 26096  Edgcedg 26160 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-edg 26161 This theorem is referenced by:  uhgredgiedgb  26242
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