Theorem edgval 29118

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 6840	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2	1	rneqd 5893	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 29117	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4		fvex 6853	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	4	rnex 7861	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
6	2, 3, 5	fvmpt 6947	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5881	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6832	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5893	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6832	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
12	8, 10, 11	3eqtr4rd 2782	. 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 2114  Vcvv 3429  ∅c0 4273  ran crn 5632  ‘cfv 6498  iEdgciedg 29066  Edgcedg 29116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fv 6506  df-edg 29117

This theorem is referenced by:  iedgedg  29119  edgopval  29120  edgstruct  29122  edgiedgb  29123  edg0iedg0  29124  uhgredgn0  29197  upgredgss  29201  umgredgss  29202  edgupgr  29203  uhgrvtxedgiedgb  29205  upgredg  29206  usgredgss  29228  ausgrumgri  29236  ausgrusgri  29237  uspgrf1oedg  29242  uspgrupgrushgr  29248  usgrumgruspgr  29251  usgruspgrb  29252  usgrf1oedg  29276  uhgr2edg  29277  usgrsizedg  29284  usgredg3  29285  ushgredgedg  29298  ushgredgedgloop  29300  usgr1e  29314  edg0usgr  29322  usgr1v0edg  29326  usgrexmpledg  29331  subgrprop3  29345  0grsubgr  29347  0uhgrsubgr  29348  subgruhgredgd  29353  uhgrspansubgrlem  29359  uhgrspan1  29372  upgrres1  29382  usgredgffibi  29393  dfnbgr3  29407  nbupgrres  29433  usgrnbcnvfv  29434  cplgrop  29506  cusgrexi  29512  structtocusgr  29515  cusgrsize  29523  1loopgredg  29570  1egrvtxdg0  29580  umgr2v2eedg  29593  edginwlk  29703  wlkl1loop  29706  wlkvtxedg  29712  uspgr2wlkeq  29714  wlkiswwlks1  29935  wlkiswwlks2lem4  29940  wlkiswwlks2lem5  29941  wlkiswwlks2  29943  wlkiswwlksupgr2  29945  2pthon3v  30011  usgrwwlks2on  30026  umgrwwlks2on  30027  clwlkclwwlk  30072  lfuhgr  35300  loop1cycl  35319  dfclnbgr3  48302  isubgredgss  48341  isubgredg  48342  isuspgrim0lem  48369  upgrimtrlslem2  48381  gricushgr  48393  ushggricedg  48403  stgredg  48432  usgrexmpl1edg  48500  usgrexmpl2edg  48505  gpgedg  48521