Theorem edgval 29066

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 6906	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2	1	rneqd 5949	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 29065	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4		fvex 6919	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	4	rnex 7932	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
6	2, 3, 5	fvmpt 7016	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5936	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6898	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5949	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6898	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
12	8, 10, 11	3eqtr4rd 2788	. 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 1540   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  ran crn 5686  ‘cfv 6561  iEdgciedg 29014  Edgcedg 29064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-edg 29065

This theorem is referenced by:  iedgedg  29067  edgopval  29068  edgstruct  29070  edgiedgb  29071  edg0iedg0  29072  uhgredgn0  29145  upgredgss  29149  umgredgss  29150  edgupgr  29151  uhgrvtxedgiedgb  29153  upgredg  29154  usgredgss  29176  ausgrumgri  29184  ausgrusgri  29185  uspgrf1oedg  29190  uspgrupgrushgr  29196  usgrumgruspgr  29199  usgruspgrb  29200  usgrf1oedg  29224  uhgr2edg  29225  usgrsizedg  29232  usgredg3  29233  ushgredgedg  29246  ushgredgedgloop  29248  usgr1e  29262  edg0usgr  29270  usgr1v0edg  29274  usgrexmpledg  29279  subgrprop3  29293  0grsubgr  29295  0uhgrsubgr  29296  subgruhgredgd  29301  uhgrspansubgrlem  29307  uhgrspan1  29320  upgrres1  29330  usgredgffibi  29341  dfnbgr3  29355  nbupgrres  29381  usgrnbcnvfv  29382  cplgrop  29454  cusgrexi  29460  structtocusgr  29463  cusgrsize  29472  1loopgredg  29519  1egrvtxdg0  29529  umgr2v2eedg  29542  edginwlk  29653  wlkl1loop  29656  wlkvtxedg  29662  uspgr2wlkeq  29664  wlkiswwlks1  29887  wlkiswwlks2lem4  29892  wlkiswwlks2lem5  29893  wlkiswwlks2  29895  wlkiswwlksupgr2  29897  2pthon3v  29963  umgrwwlks2on  29977  clwlkclwwlk  30021  lfuhgr  35123  loop1cycl  35142  dfclnbgr3  47813  isubgredgss  47851  isubgredg  47852  isuspgrim0lem  47871  gricushgr  47886  ushggricedg  47896  stgredg  47923  usgrexmpl1edg  47983  usgrexmpl2edg  47988  gpgedg  48004