Theorem edgiedgb 29072

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 29067	. . . 4 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2		edgiedgb.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	eqcomi 2745	. . . . 5 ⊢ (iEdg‘𝐺) = 𝐼
4	3	rneqi 5947	. . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼
5	1, 4	eqtri 2764	. . 3 ⊢ (Edg‘𝐺) = ran 𝐼
6	5	eleq2i 2832	. 2 ⊢ (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼)
7		elrnrexdmb 7109	. 2 ⊢ (Fun 𝐼 → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))
8	6, 7	bitrid 283	1 ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ∈ wcel 2107  ∃wrex 3069  dom cdm 5684  ran crn 5685  Fun wfun 6554  ‘cfv 6560  iEdgciedg 29015  Edgcedg 29065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-edg 29066

This theorem is referenced by:  uhgredgiedgb  29144  isubgredg  47857