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Theorem iedgedg 26063
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 6467 . 2 ((Fun 𝐸𝐼 ∈ dom 𝐸) → (𝐸𝐼) ∈ ran 𝐸)
2 edgval 26061 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
3 iedgedg.e . . . 4 𝐸 = (iEdg‘𝐺)
43rneqi 5459 . . 3 ran 𝐸 = ran (iEdg‘𝐺)
52, 4eqtr4i 2749 . 2 (Edg‘𝐺) = ran 𝐸
61, 5syl6eleqr 2814 1 ((Fun 𝐸𝐼 ∈ dom 𝐸) → (𝐸𝐼) ∈ (Edg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  dom cdm 5218  ran crn 5219  Fun wfun 5995  cfv 6001  iEdgciedg 25995  Edgcedg 26059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fn 6004  df-fv 6009  df-edg 26060
This theorem is referenced by:  edglnl  26158  numedglnl  26159
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