Theorem uhgrvtxedgiedgb 27199

Description: In a hypergraph, a vertex is incident with an edge iff it is contained in an element of the range of the edge function. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 6-Jul-2022.)

Hypotheses

Ref	Expression
uhgrvtxedgiedgb.i	⊢ 𝐼 = (iEdg‘𝐺)
uhgrvtxedgiedgb.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
uhgrvtxedgiedgb	⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖) ↔ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒))

Distinct variable groups:   𝑒,𝐸   𝑒,𝐼,𝑖   𝑈,𝑒,𝑖

Allowed substitution hints:   𝐸(𝑖)   𝐺(𝑒,𝑖)   𝑉(𝑒,𝑖)

Proof of Theorem uhgrvtxedgiedgb

Step	Hyp	Ref	Expression
1		edgval 27112	. . . . . . 7 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2	1	a1i 11	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3		uhgrvtxedgiedgb.e	. . . . . 6 ⊢ 𝐸 = (Edg‘𝐺)
4		uhgrvtxedgiedgb.i	. . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺)
5	4	rneqi 5795	. . . . . 6 ⊢ ran 𝐼 = ran (iEdg‘𝐺)
6	2, 3, 5	3eqtr4g 2799	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼)
7	6	rexeqdv 3319	. . . 4 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈 ∈ 𝑒))
8	4	uhgrfun 27129	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
9	8	funfnd 6400	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
10		eleq2 2822	. . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → (𝑈 ∈ 𝑒 ↔ 𝑈 ∈ (𝐼‘𝑖)))
11	10	rexrn 6895	. . . . 5 ⊢ (𝐼 Fn dom 𝐼 → (∃𝑒 ∈ ran 𝐼 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
12	9, 11	syl 17	. . . 4 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ ran 𝐼 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
13	7, 12	bitrd 282	. . 3 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
14	13	adantr 484	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
15	14	bicomd 226	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖) ↔ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∃wrex 3055  dom cdm 5540  ran crn 5541   Fn wfn 6364  ‘cfv 6369  iEdgciedg 27060  Edgcedg 27110  UHGraphcuhgr 27119

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 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-sbc 3688  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-fv 6377  df-edg 27111  df-uhgr 27121

This theorem is referenced by:  vtxduhgr0edgnel  27554