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

Theorem edgval 26846
 Description: The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) (Revised by AV, 8-Dec-2021.)
Assertion
Ref Expression
edgval (Edg‘𝐺) = ran (iEdg‘𝐺)

Proof of Theorem edgval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6649 . . . 4 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
21rneqd 5776 . . 3 (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3 df-edg 26845 . . 3 Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4 fvex 6662 . . . 4 (iEdg‘𝐺) ∈ V
54rnex 7603 . . 3 ran (iEdg‘𝐺) ∈ V
62, 3, 5fvmpt 6749 . 2 (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7 rn0 5764 . . . 4 ran ∅ = ∅
87a1i 11 . . 3 𝐺 ∈ V → ran ∅ = ∅)
9 fvprc 6642 . . . 4 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
109rneqd 5776 . . 3 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11 fvprc 6642 . . 3 𝐺 ∈ V → (Edg‘𝐺) = ∅)
128, 10, 113eqtr4rd 2847 . 2 𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
136, 12pm2.61i 185 1 (Edg‘𝐺) = ran (iEdg‘𝐺)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2112  Vcvv 3444  ∅c0 4246  ran crn 5524  ‘cfv 6328  iEdgciedg 26794  Edgcedg 26844 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-fv 6336  df-edg 26845 This theorem is referenced by:  iedgedg  26847  edgopval  26848  edgstruct  26850  edgiedgb  26851  edg0iedg0  26852  uhgredgn0  26925  upgredgss  26929  umgredgss  26930  edgupgr  26931  uhgrvtxedgiedgb  26933  upgredg  26934  usgredgss  26956  ausgrumgri  26964  ausgrusgri  26965  uspgrf1oedg  26970  uspgrupgrushgr  26974  usgrumgruspgr  26977  usgruspgrb  26978  usgrf1oedg  27001  uhgr2edg  27002  usgrsizedg  27009  usgredg3  27010  ushgredgedg  27023  ushgredgedgloop  27025  usgr1e  27039  edg0usgr  27047  usgr1v0edg  27051  usgrexmpledg  27056  subgrprop3  27070  0grsubgr  27072  0uhgrsubgr  27073  subgruhgredgd  27078  uhgrspansubgrlem  27084  uhgrspan1  27097  upgrres1  27107  usgredgffibi  27118  dfnbgr3  27132  nbupgrres  27158  usgrnbcnvfv  27159  cplgrop  27231  cusgrexi  27237  structtocusgr  27240  cusgrsize  27248  1loopgredg  27295  1egrvtxdg0  27305  umgr2v2eedg  27318  edginwlk  27428  wlkl1loop  27431  wlkvtxedg  27437  uspgr2wlkeq  27439  wlkiswwlks1  27657  wlkiswwlks2lem4  27662  wlkiswwlks2lem5  27663  wlkiswwlks2  27665  wlkiswwlksupgr2  27667  2pthon3v  27733  umgrwwlks2on  27747  clwlkclwwlk  27791  lfuhgr  32478  loop1cycl  32498  isomushgr  44341  ushrisomgr  44356
 Copyright terms: Public domain W3C validator