Theorem iedgedg 29141

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 29140	. . 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 2767	. 2 ⊢ (Edg‘𝐺) = ran 𝐸
6	1, 5	eleqtrrdi 2852	1 ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  dom cdm 5621  ran crn 5622  Fun wfun 6483  ‘cfv 6489  iEdgciedg 29088  Edgcedg 29138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497  df-edg 29139

This theorem is referenced by:  edglnl  29234  numedglnl  29235  umgr2cycllem  35383  uhgrimedgi  48395  isuspgrim0lem  48398  isuspgrim0  48399  upgrimwlklem2  48403  upgrimwlklem3  48404  upgrimtrlslem1  48409  clnbgrgrimlem  48438  clnbgrgrim  48439  grimedg  48440  uspgrlimlem4  48496