Theorem edgval 28340

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 6892	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2	1	rneqd 5938	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 28339	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4		fvex 6905	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	4	rnex 7903	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
6	2, 3, 5	fvmpt 6999	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5926	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6884	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5938	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6884	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
12	8, 10, 11	3eqtr4rd 2784	. 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 1542   ∈ wcel 2107  Vcvv 3475  ∅c0 4323  ran crn 5678  ‘cfv 6544  iEdgciedg 28288  Edgcedg 28338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-edg 28339

This theorem is referenced by:  iedgedg  28341  edgopval  28342  edgstruct  28344  edgiedgb  28345  edg0iedg0  28346  uhgredgn0  28419  upgredgss  28423  umgredgss  28424  edgupgr  28425  uhgrvtxedgiedgb  28427  upgredg  28428  usgredgss  28450  ausgrumgri  28458  ausgrusgri  28459  uspgrf1oedg  28464  uspgrupgrushgr  28468  usgrumgruspgr  28471  usgruspgrb  28472  usgrf1oedg  28495  uhgr2edg  28496  usgrsizedg  28503  usgredg3  28504  ushgredgedg  28517  ushgredgedgloop  28519  usgr1e  28533  edg0usgr  28541  usgr1v0edg  28545  usgrexmpledg  28550  subgrprop3  28564  0grsubgr  28566  0uhgrsubgr  28567  subgruhgredgd  28572  uhgrspansubgrlem  28578  uhgrspan1  28591  upgrres1  28601  usgredgffibi  28612  dfnbgr3  28626  nbupgrres  28652  usgrnbcnvfv  28653  cplgrop  28725  cusgrexi  28731  structtocusgr  28734  cusgrsize  28742  1loopgredg  28789  1egrvtxdg0  28799  umgr2v2eedg  28812  edginwlk  28923  wlkl1loop  28926  wlkvtxedg  28932  uspgr2wlkeq  28934  wlkiswwlks1  29152  wlkiswwlks2lem4  29157  wlkiswwlks2lem5  29158  wlkiswwlks2  29160  wlkiswwlksupgr2  29162  2pthon3v  29228  umgrwwlks2on  29242  clwlkclwwlk  29286  lfuhgr  34139  loop1cycl  34159  isomushgr  46542  ushrisomgr  46557