Theorem iedgedg 29491

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 7072	. 2 ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ ran 𝐸)
2		edgval 29490	. . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
3		iedgedg.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
4	3	rneqi 5925	. . 3 ⊢ ran 𝐸 = ran (iEdg‘𝐺)
5	2, 4	eqtr4i 2788	. 2 ⊢ (Edg‘𝐺) = ran 𝐸
6	1, 5	eleqtrrdi 2873	1 ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  dom cdm 5659  ran crn 5660  Fun wfun 6531  ‘cfv 6537  iEdgciedg 29438  Edgcedg 29488

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-edg 29489

This theorem is used by:  edglnl  29584  numedglnl  29585  umgr2cycllem  30609  uhgrimedgi  48791  isuspgrim0lem  48794  isuspgrim0  48795  upgrimwlklem2  48799  upgrimwlklem3  48800  upgrimtrlslem1  48805  clnbgrgrimlem  48834  clnbgrgrim  48835  grimedg  48836  uspgrlimlem4  48892