Theorem edgval 29020

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 6817	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2	1	rneqd 5875	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 29019	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4		fvex 6830	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	4	rnex 7835	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
6	2, 3, 5	fvmpt 6924	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5863	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6809	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5875	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6809	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
12	8, 10, 11	3eqtr4rd 2776	. 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 2110  Vcvv 3434  ∅c0 4281  ran crn 5615  ‘cfv 6477  iEdgciedg 28968  Edgcedg 29018

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7663

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6433  df-fun 6479  df-fv 6485  df-edg 29019

This theorem is referenced by:  iedgedg  29021  edgopval  29022  edgstruct  29024  edgiedgb  29025  edg0iedg0  29026  uhgredgn0  29099  upgredgss  29103  umgredgss  29104  edgupgr  29105  uhgrvtxedgiedgb  29107  upgredg  29108  usgredgss  29130  ausgrumgri  29138  ausgrusgri  29139  uspgrf1oedg  29144  uspgrupgrushgr  29150  usgrumgruspgr  29153  usgruspgrb  29154  usgrf1oedg  29178  uhgr2edg  29179  usgrsizedg  29186  usgredg3  29187  ushgredgedg  29200  ushgredgedgloop  29202  usgr1e  29216  edg0usgr  29224  usgr1v0edg  29228  usgrexmpledg  29233  subgrprop3  29247  0grsubgr  29249  0uhgrsubgr  29250  subgruhgredgd  29255  uhgrspansubgrlem  29261  uhgrspan1  29274  upgrres1  29284  usgredgffibi  29295  dfnbgr3  29309  nbupgrres  29335  usgrnbcnvfv  29336  cplgrop  29408  cusgrexi  29414  structtocusgr  29417  cusgrsize  29426  1loopgredg  29473  1egrvtxdg0  29483  umgr2v2eedg  29496  edginwlk  29606  wlkl1loop  29609  wlkvtxedg  29615  uspgr2wlkeq  29617  wlkiswwlks1  29838  wlkiswwlks2lem4  29843  wlkiswwlks2lem5  29844  wlkiswwlks2  29846  wlkiswwlksupgr2  29848  2pthon3v  29914  umgrwwlks2on  29928  clwlkclwwlk  29972  lfuhgr  35130  loop1cycl  35149  dfclnbgr3  47836  isubgredgss  47875  isubgredg  47876  isuspgrim0lem  47903  upgrimtrlslem2  47915  gricushgr  47927  ushggricedg  47937  stgredg  47966  usgrexmpl1edg  48034  usgrexmpl2edg  48039  gpgedg  48055