MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  edgvalOLD Structured version   Visualization version   GIF version

Theorem edgvalOLD 26162
Description: Obsolete version of edgval 26161 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 26160 . . 3 Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
21a1i 11 . 2 (𝐺𝑉 → Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔)))
3 fveq2 6353 . . . 4 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
43rneqd 5508 . . 3 (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
54adantl 473 . 2 ((𝐺𝑉𝑔 = 𝐺) → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
6 elex 3352 . 2 (𝐺𝑉𝐺 ∈ V)
7 fvex 6363 . . . 4 (iEdg‘𝐺) ∈ V
87rnex 7266 . . 3 ran (iEdg‘𝐺) ∈ V
98a1i 11 . 2 (𝐺𝑉 → ran (iEdg‘𝐺) ∈ V)
102, 5, 6, 9fvmptd 6451 1 (𝐺𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cmpt 4881  ran crn 5267  cfv 6049  iEdgciedg 26095  Edgcedg 26159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-edg 26160
This theorem is referenced by:  edgiedgbOLD  26168  edg0iedg0OLD  26170  edginwlkOLD  26762
  Copyright terms: Public domain W3C validator