Theorem iedgedg 29029

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 7066	. 2 ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ ran 𝐸)
2		edgval 29028	. . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3		iedgedg.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
4	3	rneqi 5917	. . 3 ⊢ ran 𝐸 = ran (iEdg‘𝐺)
5	2, 4	eqtr4i 2761	. 2 ⊢ (Edg‘𝐺) = ran 𝐸
6	1, 5	eleqtrrdi 2845	1 ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  dom cdm 5654  ran crn 5655  Fun wfun 6525  ‘cfv 6531  iEdgciedg 28976  Edgcedg 29026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fn 6534  df-fv 6539  df-edg 29027

This theorem is referenced by:  edglnl  29122  numedglnl  29123  umgr2cycllem  35162  uhgrimedgi  47903  isuspgrim0lem  47906  isuspgrim0  47907  upgrimwlklem2  47911  upgrimwlklem3  47912  upgrimtrlslem1  47917  clnbgrgrimlem  47946  clnbgrgrim  47947  grimedg  47948  uspgrlimlem4  48003