Theorem iedgedg 29125

Description: An indexed edge is an edge. (Contributed by AV, 19-Dec-2021.)

Hypothesis

Ref	Expression
iedgedg.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
iedgedg	⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺))

Proof of Theorem iedgedg

Step	Hyp	Ref	Expression
1		fvelrn 7021	. 2 ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ ran 𝐸)
2		edgval 29124	. . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3		iedgedg.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
4	3	rneqi 5886	. . 3 ⊢ ran 𝐸 = ran (iEdg‘𝐺)
5	2, 4	eqtr4i 2762	. 2 ⊢ (Edg‘𝐺) = ran 𝐸
6	1, 5	eleqtrrdi 2847	1 ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  dom cdm 5624  ran crn 5625  Fun wfun 6486  ‘cfv 6492  iEdgciedg 29072  Edgcedg 29122

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  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-fn 6495  df-fv 6500  df-edg 29123

This theorem is referenced by:  edglnl  29218  numedglnl  29219  umgr2cycllem  35336  uhgrimedgi  48157  isuspgrim0lem  48160  isuspgrim0  48161  upgrimwlklem2  48165  upgrimwlklem3  48166  upgrimtrlslem1  48171  clnbgrgrimlem  48200  clnbgrgrim  48201  grimedg  48202  uspgrlimlem4  48258