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Theorem edgvalOLD 25987
Description: Obsolete version of edgval 25986 as of 8-Dec-2021. (Contributed by AV, 1-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
edgvalOLD (𝐺𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))

Proof of Theorem edgvalOLD
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 df-edg 25985 . . 3 Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
21a1i 11 . 2 (𝐺𝑉 → Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔)))
3 fveq2 6229 . . . 4 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
43rneqd 5385 . . 3 (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
54adantl 481 . 2 ((𝐺𝑉𝑔 = 𝐺) → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
6 elex 3243 . 2 (𝐺𝑉𝐺 ∈ V)
7 fvex 6239 . . . 4 (iEdg‘𝐺) ∈ V
87rnex 7142 . . 3 ran (iEdg‘𝐺) ∈ V
98a1i 11 . 2 (𝐺𝑉 → ran (iEdg‘𝐺) ∈ V)
102, 5, 6, 9fvmptd 6327 1 (𝐺𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cmpt 4762  ran crn 5144  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-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-fv 5934  df-edg 25985
This theorem is referenced by:  edgiedgbOLD  25993  edg0iedg0OLD  25995  edginwlkOLD  26587
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