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

Theorem edgaval 25794
 Description: The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
Assertion
Ref Expression
edgaval (𝐺𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))

Proof of Theorem edgaval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 df-edga 25793 . . 3 Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
21a1i 11 . 2 (𝐺𝑉 → Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔)))
3 fveq2 6103 . . . 4 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
43rneqd 5274 . . 3 (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
54adantl 481 . 2 ((𝐺𝑉𝑔 = 𝐺) → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
6 elex 3185 . 2 (𝐺𝑉𝐺 ∈ V)
7 fvex 6113 . . . 4 (iEdg‘𝐺) ∈ V
87rnex 6992 . . 3 ran (iEdg‘𝐺) ∈ V
98a1i 11 . 2 (𝐺𝑉 → ran (iEdg‘𝐺) ∈ V)
102, 5, 6, 9fvmptd 6197 1 (𝐺𝑉 → (Edg‘𝐺) = ran (iEdg‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  iEdgciedg 25674  Edgcedga 25792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-edga 25793 This theorem is referenced by:  edgaopval  25795  edgastruct  25797  edgiedgb  25798  edg0iedg0  25800  uhgredgn0  25802  upgredgss  25806  umgredgss  25807  edgupgr  25808  uhgrvtxedgiedgb  25810  upgredg  25811  usgredgss  40390  ausgrumgri  40397  ausgrusgri  40398  uspgrf1oedg  40403  uspgrupgrushgr  40407  usgrumgruspgr  40410  usgruspgrb  40411  usgrf1oedg  40434  uhgr2edg  40435  usgrsizedg  40442  usgredg3  40443  ushgredgedga  40456  ushgredgedgaloop  40458  usgr1e  40471  edg0usgr  40479  usgr1v0edg  40483  usgrexmpledg  40486  subgrprop3  40500  0grsubgr  40502  0uhgrsubgr  40503  subgruhgredgd  40508  uhgrspansubgrlem  40514  uhgrspan1  40527  upgrres1  40532  usgredgffibi  40543  dfnbgr3  40562  nbupgrres  40592  usgrnbcnvfv  40593  cplgrop  40659  cusgrexi  40662  cusgrsize  40670  1loopgredg  40716  1egrvtxdg0  40727  umgr2v2eedg  40740  edginwlk  40839  1wlkl1loop  40842  1wlkvtxedg  40852  uspgr2wlkeq  40854  1wlkiswwlks1  41064  1wlkiswwlks2lem4  41069  1wlkiswwlks2lem5  41070  1wlkiswwlks2  41072  1wlkiswwlksupgr2  41074  2pthon3v-av  41150  umgrwwlks2on  41161  clwlkclwwlk  41211  clwlksfclwwlk  41269
 Copyright terms: Public domain W3C validator