Theorem edgiedgb 25798

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

Hypothesis

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

Assertion

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

Distinct variable groups:   𝑥,𝐸   𝑥,𝐼

Allowed substitution hints:   𝐺(𝑥)   𝑊(𝑥)

Proof of Theorem edgiedgb

Step	Hyp	Ref	Expression
1		edgaval 25794	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (Edg‘𝐺) = ran (iEdg‘𝐺))
2	1	adantr 480	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (Edg‘𝐺) = ran (iEdg‘𝐺))
3		edgiedgb.i	. . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺)
4	3	eqcomi 2619	. . . . . 6 ⊢ (iEdg‘𝐺) = 𝐼
5	4	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (iEdg‘𝐺) = 𝐼)
6	5	rneqd 5274	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → ran (iEdg‘𝐺) = ran 𝐼)
7	2, 6	eqtrd 2644	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (Edg‘𝐺) = ran 𝐼)
8	7	eleq2d 2673	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼))
9		elrnrexdmb 6272	. . 3 ⊢ (Fun 𝐼 → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))
10	9	adantl 481	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))
11	8, 10	bitrd 267	1 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐼) → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  dom cdm 5038  ran crn 5039  Fun wfun 5798  ‘cfv 5804  iEdgciedg 25674  Edgcedga 25792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-edga 25793

This theorem is referenced by:  uhgredgiedgb  25799