Theorem edgval 29029

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 6828	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2	1	rneqd 5882	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 29028	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4		fvex 6841	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	4	rnex 7846	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
6	2, 3, 5	fvmpt 6935	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5870	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6820	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5882	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6820	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
12	8, 10, 11	3eqtr4rd 2779	. 2 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
13	6, 12	pm2.61i 182	1 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282  ran crn 5620  ‘cfv 6486  iEdgciedg 28977  Edgcedg 29027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fv 6494  df-edg 29028

This theorem is referenced by:  iedgedg  29030  edgopval  29031  edgstruct  29033  edgiedgb  29034  edg0iedg0  29035  uhgredgn0  29108  upgredgss  29112  umgredgss  29113  edgupgr  29114  uhgrvtxedgiedgb  29116  upgredg  29117  usgredgss  29139  ausgrumgri  29147  ausgrusgri  29148  uspgrf1oedg  29153  uspgrupgrushgr  29159  usgrumgruspgr  29162  usgruspgrb  29163  usgrf1oedg  29187  uhgr2edg  29188  usgrsizedg  29195  usgredg3  29196  ushgredgedg  29209  ushgredgedgloop  29211  usgr1e  29225  edg0usgr  29233  usgr1v0edg  29237  usgrexmpledg  29242  subgrprop3  29256  0grsubgr  29258  0uhgrsubgr  29259  subgruhgredgd  29264  uhgrspansubgrlem  29270  uhgrspan1  29283  upgrres1  29293  usgredgffibi  29304  dfnbgr3  29318  nbupgrres  29344  usgrnbcnvfv  29345  cplgrop  29417  cusgrexi  29423  structtocusgr  29426  cusgrsize  29435  1loopgredg  29482  1egrvtxdg0  29492  umgr2v2eedg  29505  edginwlk  29615  wlkl1loop  29618  wlkvtxedg  29624  uspgr2wlkeq  29626  wlkiswwlks1  29847  wlkiswwlks2lem4  29852  wlkiswwlks2lem5  29853  wlkiswwlks2  29855  wlkiswwlksupgr2  29857  2pthon3v  29923  usgrwwlks2on  29938  umgrwwlks2on  29939  clwlkclwwlk  29984  lfuhgr  35183  loop1cycl  35202  dfclnbgr3  47950  isubgredgss  47989  isubgredg  47990  isuspgrim0lem  48017  upgrimtrlslem2  48029  gricushgr  48041  ushggricedg  48051  stgredg  48080  usgrexmpl1edg  48148  usgrexmpl2edg  48153  gpgedg  48169