Theorem edgval 28976

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 6858	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2	1	rneqd 5902	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 28975	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4		fvex 6871	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	4	rnex 7886	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
6	2, 3, 5	fvmpt 6968	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5889	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6850	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5902	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6850	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
12	8, 10, 11	3eqtr4rd 2775	. 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 2109  Vcvv 3447  ∅c0 4296  ran crn 5639  ‘cfv 6511  iEdgciedg 28924  Edgcedg 28974

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-edg 28975

This theorem is referenced by:  iedgedg  28977  edgopval  28978  edgstruct  28980  edgiedgb  28981  edg0iedg0  28982  uhgredgn0  29055  upgredgss  29059  umgredgss  29060  edgupgr  29061  uhgrvtxedgiedgb  29063  upgredg  29064  usgredgss  29086  ausgrumgri  29094  ausgrusgri  29095  uspgrf1oedg  29100  uspgrupgrushgr  29106  usgrumgruspgr  29109  usgruspgrb  29110  usgrf1oedg  29134  uhgr2edg  29135  usgrsizedg  29142  usgredg3  29143  ushgredgedg  29156  ushgredgedgloop  29158  usgr1e  29172  edg0usgr  29180  usgr1v0edg  29184  usgrexmpledg  29189  subgrprop3  29203  0grsubgr  29205  0uhgrsubgr  29206  subgruhgredgd  29211  uhgrspansubgrlem  29217  uhgrspan1  29230  upgrres1  29240  usgredgffibi  29251  dfnbgr3  29265  nbupgrres  29291  usgrnbcnvfv  29292  cplgrop  29364  cusgrexi  29370  structtocusgr  29373  cusgrsize  29382  1loopgredg  29429  1egrvtxdg0  29439  umgr2v2eedg  29452  edginwlk  29563  wlkl1loop  29566  wlkvtxedg  29572  uspgr2wlkeq  29574  wlkiswwlks1  29797  wlkiswwlks2lem4  29802  wlkiswwlks2lem5  29803  wlkiswwlks2  29805  wlkiswwlksupgr2  29807  2pthon3v  29873  umgrwwlks2on  29887  clwlkclwwlk  29931  lfuhgr  35105  loop1cycl  35124  dfclnbgr3  47827  isubgredgss  47865  isubgredg  47866  isuspgrim0lem  47893  upgrimtrlslem2  47905  gricushgr  47917  ushggricedg  47927  stgredg  47955  usgrexmpl1edg  48015  usgrexmpl2edg  48020  gpgedg  48036