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Theorem iedgedg 25988
 Description: An indexed edge is an edge. (Contributed by AV, 19-Dec-2021.)
Hypothesis
Ref Expression
iedgedg.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
iedgedg ((Fun 𝐸𝐼 ∈ dom 𝐸) → (𝐸𝐼) ∈ (Edg‘𝐺))

Proof of Theorem iedgedg
StepHypRef Expression
1 fvelrn 6392 . 2 ((Fun 𝐸𝐼 ∈ dom 𝐸) → (𝐸𝐼) ∈ ran 𝐸)
2 edgval 25986 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
3 iedgedg.e . . . 4 𝐸 = (iEdg‘𝐺)
43rneqi 5384 . . 3 ran 𝐸 = ran (iEdg‘𝐺)
52, 4eqtr4i 2676 . 2 (Edg‘𝐺) = ran 𝐸
61, 5syl6eleqr 2741 1 ((Fun 𝐸𝐼 ∈ dom 𝐸) → (𝐸𝐼) ∈ (Edg‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  dom cdm 5143  ran crn 5144  Fun wfun 5920  ‘cfv 5926  iEdgciedg 25920  Edgcedg 25984 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-edg 25985 This theorem is referenced by:  edglnl  26083  numedglnl  26084
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