Theorem edgval 29132

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 6834	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2	1	rneqd 5887	. . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺))
3		df-edg 29131	. . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔))
4		fvex 6847	. . . 4 ⊢ (iEdg‘𝐺) ∈ V
5	4	rnex 7854	. . 3 ⊢ ran (iEdg‘𝐺) ∈ V
6	2, 3, 5	fvmpt 6941	. 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
7		rn0 5875	. . . 4 ⊢ ran ∅ = ∅
8	7	a1i 11	. . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅)
9		fvprc 6826	. . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅)
10	9	rneqd 5887	. . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅)
11		fvprc 6826	. . 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 3430  ∅c0 4274  ran crn 5625  ‘cfv 6492  iEdgciedg 29080  Edgcedg 29130

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 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-edg 29131

This theorem is referenced by:  iedgedg  29133  edgopval  29134  edgstruct  29136  edgiedgb  29137  edg0iedg0  29138  uhgredgn0  29211  upgredgss  29215  umgredgss  29216  edgupgr  29217  uhgrvtxedgiedgb  29219  upgredg  29220  usgredgss  29242  ausgrumgri  29250  ausgrusgri  29251  uspgrf1oedg  29256  uspgrupgrushgr  29262  usgrumgruspgr  29265  usgruspgrb  29266  usgrf1oedg  29290  uhgr2edg  29291  usgrsizedg  29298  usgredg3  29299  ushgredgedg  29312  ushgredgedgloop  29314  usgr1e  29328  edg0usgr  29336  usgr1v0edg  29340  usgrexmpledg  29345  subgrprop3  29359  0grsubgr  29361  0uhgrsubgr  29362  subgruhgredgd  29367  uhgrspansubgrlem  29373  uhgrspan1  29386  upgrres1  29396  usgredgffibi  29407  dfnbgr3  29421  nbupgrres  29447  usgrnbcnvfv  29448  cplgrop  29520  cusgrexi  29526  structtocusgr  29529  cusgrsize  29538  1loopgredg  29585  1egrvtxdg0  29595  umgr2v2eedg  29608  edginwlk  29718  wlkl1loop  29721  wlkvtxedg  29727  uspgr2wlkeq  29729  wlkiswwlks1  29950  wlkiswwlks2lem4  29955  wlkiswwlks2lem5  29956  wlkiswwlks2  29958  wlkiswwlksupgr2  29960  2pthon3v  30026  usgrwwlks2on  30041  umgrwwlks2on  30042  clwlkclwwlk  30087  lfuhgr  35316  loop1cycl  35335  dfclnbgr3  48314  isubgredgss  48353  isubgredg  48354  isuspgrim0lem  48381  upgrimtrlslem2  48393  gricushgr  48405  ushggricedg  48415  stgredg  48444  usgrexmpl1edg  48512  usgrexmpl2edg  48517  gpgedg  48533