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Theorem iedgedg 26846
 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 6825 . 2 ((Fun 𝐸𝐼 ∈ dom 𝐸) → (𝐸𝐼) ∈ ran 𝐸)
2 edgval 26845 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
3 iedgedg.e . . . 4 𝐸 = (iEdg‘𝐺)
43rneqi 5775 . . 3 ran 𝐸 = ran (iEdg‘𝐺)
52, 4eqtr4i 2827 . 2 (Edg‘𝐺) = ran 𝐸
61, 5eleqtrrdi 2904 1 ((Fun 𝐸𝐼 ∈ dom 𝐸) → (𝐸𝐼) ∈ (Edg‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  dom cdm 5523  ran crn 5524  Fun wfun 6322  ‘cfv 6328  iEdgciedg 26793  Edgcedg 26843 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-edg 26844 This theorem is referenced by:  edglnl  26939  numedglnl  26940  umgr2cycllem  32495
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