Theorem edgval 29134

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 6842	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2	1	rneqd 5895	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 29133	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4		fvex 6855	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	4	rnex 7862	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
6	2, 3, 5	fvmpt 6949	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5883	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6834	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5895	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6834	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
12	8, 10, 11	3eqtr4rd 2783	. 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 3442  ∅c0 4287  ran crn 5633  ‘cfv 6500  iEdgciedg 29082  Edgcedg 29132

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 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fv 6508  df-edg 29133

This theorem is referenced by:  iedgedg  29135  edgopval  29136  edgstruct  29138  edgiedgb  29139  edg0iedg0  29140  uhgredgn0  29213  upgredgss  29217  umgredgss  29218  edgupgr  29219  uhgrvtxedgiedgb  29221  upgredg  29222  usgredgss  29244  ausgrumgri  29252  ausgrusgri  29253  uspgrf1oedg  29258  uspgrupgrushgr  29264  usgrumgruspgr  29267  usgruspgrb  29268  usgrf1oedg  29292  uhgr2edg  29293  usgrsizedg  29300  usgredg3  29301  ushgredgedg  29314  ushgredgedgloop  29316  usgr1e  29330  edg0usgr  29338  usgr1v0edg  29342  usgrexmpledg  29347  subgrprop3  29361  0grsubgr  29363  0uhgrsubgr  29364  subgruhgredgd  29369  uhgrspansubgrlem  29375  uhgrspan1  29388  upgrres1  29398  usgredgffibi  29409  dfnbgr3  29423  nbupgrres  29449  usgrnbcnvfv  29450  cplgrop  29522  cusgrexi  29528  structtocusgr  29531  cusgrsize  29540  1loopgredg  29587  1egrvtxdg0  29597  umgr2v2eedg  29610  edginwlk  29720  wlkl1loop  29723  wlkvtxedg  29729  uspgr2wlkeq  29731  wlkiswwlks1  29952  wlkiswwlks2lem4  29957  wlkiswwlks2lem5  29958  wlkiswwlks2  29960  wlkiswwlksupgr2  29962  2pthon3v  30028  usgrwwlks2on  30043  umgrwwlks2on  30044  clwlkclwwlk  30089  lfuhgr  35331  loop1cycl  35350  dfclnbgr3  48180  isubgredgss  48219  isubgredg  48220  isuspgrim0lem  48247  upgrimtrlslem2  48259  gricushgr  48271  ushggricedg  48281  stgredg  48310  usgrexmpl1edg  48378  usgrexmpl2edg  48383  gpgedg  48399