Theorem uhgrvtxedgiedgb 29063

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 28976	. . . . . . 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 5901	. . . . . 6 ⊢ ran 𝐼 = ran (iEdg‘𝐺)
6	2, 3, 5	3eqtr4g 2789	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼)
7	6	rexeqdv 3300	. . . 4 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈 ∈ 𝑒))
8	4	uhgrfun 28993	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
9	8	funfnd 6547	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
10		eleq2 2817	. . . . . 6 ⊢ (𝑒 = (𝐼‘𝑖) → (𝑈 ∈ 𝑒 ↔ 𝑈 ∈ (𝐼‘𝑖)))
11	10	rexrn 7059	. . . . 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 279	. . 3 ⊢ (𝐺 ∈ UHGraph → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
14	13	adantr 480	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖)))
15	14	bicomd 223	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑈 ∈ 𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼‘𝑖) ↔ ∃𝑒 ∈ 𝐸 𝑈 ∈ 𝑒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  dom cdm 5638  ran crn 5639   Fn wfn 6506  ‘cfv 6511  iEdgciedg 28924  Edgcedg 28974  UHGraphcuhgr 28983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-edg 28975  df-uhgr 28985

This theorem is referenced by:  vtxduhgr0edgnel  29422