Theorem edgval 25986

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		df-edg 25985	. . . 4 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
2	1	a1i 11	. . 3 ⊢ (𝐺 ∈ V → Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔)))
3		fveq2 6229	. . . . 5 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
4	3	rneqd 5385	. . . 4 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
5	4	adantl 481	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑔 = 𝐺) → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
6		id 22	. . 3 ⊢ (𝐺 ∈ V → 𝐺 ∈ V)
7		fvex 6239	. . . . 5 ⊢ (iEdg‘𝐺) ∈ V
8	7	rnex 7142	. . . 4 ⊢ ran (iEdg‘𝐺) ∈ V
9	8	a1i 11	. . 3 ⊢ (𝐺 ∈ V → ran (iEdg‘𝐺) ∈ V)
10	2, 5, 6, 9	fvmptd 6327	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
11		rn0 5409	. . . 4 ⊢ ran ∅ = ∅
12	11	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
13		fvprc 6223	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
14	13	rneqd 5385	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
15		fvprc 6223	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
16	12, 14, 15	3eqtr4rd 2696	. 2 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
17	10, 16	pm2.61i 176	1 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)

Syntax hints:  ¬ wn 3   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ∅c0 3948   ↦ cmpt 4762  ran crn 5144  ‘cfv 5926  iEdgciedg 25920  Edgcedg 25984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-edg 25985

This theorem is referenced by:  iedgedg  25988  edgopval  25989  edgstruct  25991  edgiedgb  25992  edg0iedg0  25994  uhgredgn0  26068  upgredgss  26072  umgredgss  26073  edgupgr  26074  uhgrvtxedgiedgb  26076  upgredg  26077  usgredgss  26099  ausgrumgri  26107  ausgrusgri  26108  uspgrf1oedg  26113  uspgrupgrushgr  26117  usgrumgruspgr  26120  usgruspgrb  26121  usgrf1oedg  26144  uhgr2edg  26145  usgrsizedg  26152  usgredg3  26153  ushgredgedg  26166  ushgredgedgloop  26168  usgr1e  26182  edg0usgr  26190  usgr1v0edg  26194  usgrexmpledg  26199  subgrprop3  26213  0grsubgr  26215  0uhgrsubgr  26216  subgruhgredgd  26221  uhgrspansubgrlem  26227  uhgrspan1  26240  upgrres1  26250  usgredgffibi  26261  dfnbgr3  26276  nbupgrres  26310  usgrnbcnvfv  26311  cplgrop  26389  cusgrexi  26395  structtocusgr  26398  cusgrsize  26406  1loopgredg  26453  1egrvtxdg0  26463  umgr2v2eedg  26476  edginwlk  26586  wlkl1loop  26590  wlkvtxedg  26596  uspgr2wlkeq  26598  wlkiswwlks1  26821  wlkiswwlks2lem4  26826  wlkiswwlks2lem5  26827  wlkiswwlks2  26829  wlkiswwlksupgr2  26831  2pthon3v  26908  umgrwwlks2on  26923  clwlkclwwlk  26968  clwlksfclwwlk  27049