Theorem edgval 26834

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 6670	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2	1	rneqd 5808	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 26833	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4		fvex 6683	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	4	rnex 7617	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
6	2, 3, 5	fvmpt 6768	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5796	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6663	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5808	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6663	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
12	8, 10, 11	3eqtr4rd 2867	. 2 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
13	6, 12	pm2.61i 184	1 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ∅c0 4291  ran crn 5556  ‘cfv 6355  iEdgciedg 26782  Edgcedg 26832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fv 6363  df-edg 26833

This theorem is referenced by:  iedgedg  26835  edgopval  26836  edgstruct  26838  edgiedgb  26839  edg0iedg0  26840  uhgredgn0  26913  upgredgss  26917  umgredgss  26918  edgupgr  26919  uhgrvtxedgiedgb  26921  upgredg  26922  usgredgss  26944  ausgrumgri  26952  ausgrusgri  26953  uspgrf1oedg  26958  uspgrupgrushgr  26962  usgrumgruspgr  26965  usgruspgrb  26966  usgrf1oedg  26989  uhgr2edg  26990  usgrsizedg  26997  usgredg3  26998  ushgredgedg  27011  ushgredgedgloop  27013  usgr1e  27027  edg0usgr  27035  usgr1v0edg  27039  usgrexmpledg  27044  subgrprop3  27058  0grsubgr  27060  0uhgrsubgr  27061  subgruhgredgd  27066  uhgrspansubgrlem  27072  uhgrspan1  27085  upgrres1  27095  usgredgffibi  27106  dfnbgr3  27120  nbupgrres  27146  usgrnbcnvfv  27147  cplgrop  27219  cusgrexi  27225  structtocusgr  27228  cusgrsize  27236  1loopgredg  27283  1egrvtxdg0  27293  umgr2v2eedg  27306  edginwlk  27416  wlkl1loop  27419  wlkvtxedg  27425  uspgr2wlkeq  27427  wlkiswwlks1  27645  wlkiswwlks2lem4  27650  wlkiswwlks2lem5  27651  wlkiswwlks2  27653  wlkiswwlksupgr2  27655  2pthon3v  27722  umgrwwlks2on  27736  clwlkclwwlk  27780  lfuhgr  32364  loop1cycl  32384  isomushgr  44040  ushrisomgr  44055