Theorem edgval 29377

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.

Step	Hyp	Ref	Expression
1		fveq2 6883	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2	1	rneqd 5930	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 29376	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4		fvex 6896	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	4	rnex 7908	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
6	2, 3, 5	fvmpt 6991	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5918	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6875	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5930	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6875	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
12	8, 10, 11	3eqtr4rd 2809	. 2 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
13	6, 12	pm2.61i 184	1 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4287  ran crn 5664  ‘cfv 6538  iEdgciedg 29325  Edgcedg 29375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fv 6546  df-edg 29376

This theorem is referenced by:  iedgedg  29378  edgopval  29379  edgstruct  29381  edgiedgb  29382  edg0iedg0  29383  uhgredgn0  29456  upgredgss  29460  umgredgss  29461  edgupgr  29462  uhgrvtxedgiedgb  29464  upgredg  29465  usgredgss  29487  ausgrumgri  29495  ausgrusgri  29496  uspgrf1oedg  29501  uspgrupgrushgr  29507  usgrumgruspgr  29510  usgruspgrb  29511  usgrf1oedg  29535  uhgr2edg  29536  usgrsizedg  29543  usgredg3  29544  ushgredgedg  29557  ushgredgedgloop  29559  usgr1e  29573  edg0usgr  29581  usgr1v0edg  29585  usgrexmpledg  29590  subgrprop3  29604  0grsubgr  29606  0uhgrsubgr  29607  subgruhgredgd  29612  uhgrspansubgrlem  29618  uhgrspan1  29631  upgrres1  29641  usgredgffibi  29652  dfnbgr3  29666  nbupgrres  29692  usgrnbcnvfv  29693  cplgrop  29765  cusgrexi  29771  structtocusgr  29774  cusgrsize  29782  1loopgredg  29829  1egrvtxdg0  29839  umgr2v2eedg  29852  edginwlk  29962  wlkl1loop  29965  wlkvtxedg  29971  uspgr2wlkeq  29973  wlkiswwlks1  30194  wlkiswwlks2lem4  30199  wlkiswwlks2lem5  30200  wlkiswwlks2  30202  wlkiswwlksupgr2  30204  2pthon3v  30270  usgrwwlks2on  30285  umgrwwlks2on  30286  clwlkclwwlk  30331  lfuhgr  35588  loop1cycl  35607  dfclnbgr3  48568  isubgredgss  48607  isubgredg  48608  isuspgrim0lem  48635  upgrimtrlslem2  48647  gricushgr  48659  ushggricedg  48669  stgredg  48698  usgrexmpl1edg  48766  usgrexmpl2edg  48771  gpgedg  48787