Theorem edgval 29080

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 5951	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 29079	. . 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 7015	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5938	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6898	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5951	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6898	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
12	8, 10, 11	3eqtr4rd 2785	. 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 1536   ∈ wcel 2105  Vcvv 3477  ∅c0 4338  ran crn 5689  ‘cfv 6562  iEdgciedg 29028  Edgcedg 29078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fv 6570  df-edg 29079

This theorem is referenced by:  iedgedg  29081  edgopval  29082  edgstruct  29084  edgiedgb  29085  edg0iedg0  29086  uhgredgn0  29159  upgredgss  29163  umgredgss  29164  edgupgr  29165  uhgrvtxedgiedgb  29167  upgredg  29168  usgredgss  29190  ausgrumgri  29198  ausgrusgri  29199  uspgrf1oedg  29204  uspgrupgrushgr  29210  usgrumgruspgr  29213  usgruspgrb  29214  usgrf1oedg  29238  uhgr2edg  29239  usgrsizedg  29246  usgredg3  29247  ushgredgedg  29260  ushgredgedgloop  29262  usgr1e  29276  edg0usgr  29284  usgr1v0edg  29288  usgrexmpledg  29293  subgrprop3  29307  0grsubgr  29309  0uhgrsubgr  29310  subgruhgredgd  29315  uhgrspansubgrlem  29321  uhgrspan1  29334  upgrres1  29344  usgredgffibi  29355  dfnbgr3  29369  nbupgrres  29395  usgrnbcnvfv  29396  cplgrop  29468  cusgrexi  29474  structtocusgr  29477  cusgrsize  29486  1loopgredg  29533  1egrvtxdg0  29543  umgr2v2eedg  29556  edginwlk  29667  wlkl1loop  29670  wlkvtxedg  29676  uspgr2wlkeq  29678  wlkiswwlks1  29896  wlkiswwlks2lem4  29901  wlkiswwlks2lem5  29902  wlkiswwlks2  29904  wlkiswwlksupgr2  29906  2pthon3v  29972  umgrwwlks2on  29986  clwlkclwwlk  30030  lfuhgr  35101  loop1cycl  35121  dfclnbgr3  47750  isubgredgss  47788  isubgredg  47789  isuspgrim0lem  47808  gricushgr  47823  ushggricedg  47833  stgredg  47858  usgrexmpl1edg  47918  usgrexmpl2edg  47923  gpgedg  47939