Theorem edgval 26836

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 6672	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2	1	rneqd 5810	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 26835	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4		fvex 6685	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	4	rnex 7619	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
6	2, 3, 5	fvmpt 6770	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5798	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6665	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5810	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6665	. . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅)
12	8, 10, 11	3eqtr4rd 2869	. 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 3496  ∅c0 4293  ran crn 5558  ‘cfv 6357  iEdgciedg 26784  Edgcedg 26834

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 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fv 6365  df-edg 26835

This theorem is referenced by:  iedgedg  26837  edgopval  26838  edgstruct  26840  edgiedgb  26841  edg0iedg0  26842  uhgredgn0  26915  upgredgss  26919  umgredgss  26920  edgupgr  26921  uhgrvtxedgiedgb  26923  upgredg  26924  usgredgss  26946  ausgrumgri  26954  ausgrusgri  26955  uspgrf1oedg  26960  uspgrupgrushgr  26964  usgrumgruspgr  26967  usgruspgrb  26968  usgrf1oedg  26991  uhgr2edg  26992  usgrsizedg  26999  usgredg3  27000  ushgredgedg  27013  ushgredgedgloop  27015  usgr1e  27029  edg0usgr  27037  usgr1v0edg  27041  usgrexmpledg  27046  subgrprop3  27060  0grsubgr  27062  0uhgrsubgr  27063  subgruhgredgd  27068  uhgrspansubgrlem  27074  uhgrspan1  27087  upgrres1  27097  usgredgffibi  27108  dfnbgr3  27122  nbupgrres  27148  usgrnbcnvfv  27149  cplgrop  27221  cusgrexi  27227  structtocusgr  27230  cusgrsize  27238  1loopgredg  27285  1egrvtxdg0  27295  umgr2v2eedg  27308  edginwlk  27418  wlkl1loop  27421  wlkvtxedg  27427  uspgr2wlkeq  27429  wlkiswwlks1  27647  wlkiswwlks2lem4  27652  wlkiswwlks2lem5  27653  wlkiswwlks2  27655  wlkiswwlksupgr2  27657  2pthon3v  27724  umgrwwlks2on  27738  clwlkclwwlk  27782  lfuhgr  32366  loop1cycl  32386  isomushgr  43998  ushrisomgr  44013