Theorem edgiedgb 29257

Description: A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.) (Revised by AV, 8-Dec-2021.)

Hypothesis

Ref	Expression
edgiedgb.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
edgiedgb	⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))

Distinct variable groups:   𝑥,𝐸   𝑥,𝐼

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem edgiedgb

Step	Hyp	Ref	Expression
1		edgval 29252	. . . 4 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2		edgiedgb.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	eqcomi 2773	. . . . 5 ⊢ (iEdg‘𝐺) = 𝐼
4	3	rneqi 5915	. . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼
5	1, 4	eqtri 2787	. . 3 ⊢ (Edg‘𝐺) = ran 𝐼
6	5	eleq2i 2856	. 2 ⊢ (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼)
7		elrnrexdmb 7073	. 2 ⊢ (Fun 𝐼 → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))
8	6, 7	bitrid 285	1 ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1562   ∈ wcel 2144  ∃wrex 3088  dom cdm 5649  ran crn 5650  Fun wfun 6517  ‘cfv 6523  iEdgciedg 29200  Edgcedg 29250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-iota 6479  df-fun 6525  df-fn 6526  df-fv 6531  df-edg 29251

This theorem is referenced by:  uhgredgiedgb  29329  isubgredg  48493  uhgrimedgi  48517