Theorem uhgrauhgr 40376

Description: Equivalence of the definition for undirected hypergraphs. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)

Assertion

Ref	Expression
uhgrauhgr	⊢ ((𝑉 UHGrph 𝐸 ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝐺 ∈ 𝑊 → 𝐺 ∈ UHGraph ))

Proof of Theorem uhgrauhgr

Step	Hyp	Ref	Expression
1		uhgraf 25828	. . . 4 ⊢ (𝑉 UHGrph 𝐸 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
2	1	3ad2ant1 1075	. . 3 ⊢ ((𝑉 UHGrph 𝐸 ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
3		simp3 1056	. . . 4 ⊢ ((𝑉 UHGrph 𝐸 ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → 𝐸 = (iEdg‘𝐺))
4	3	dmeqd 5248	. . . 4 ⊢ ((𝑉 UHGrph 𝐸 ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → dom 𝐸 = dom (iEdg‘𝐺))
5		pweq 4111	. . . . . 6 ⊢ (𝑉 = (Vtx‘𝐺) → 𝒫 𝑉 = 𝒫 (Vtx‘𝐺))
6	5	difeq1d 3689	. . . . 5 ⊢ (𝑉 = (Vtx‘𝐺) → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐺) ∖ {∅}))
7	6	3ad2ant2 1076	. . . 4 ⊢ ((𝑉 UHGrph 𝐸 ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝒫 𝑉 ∖ {∅}) = (𝒫 (Vtx‘𝐺) ∖ {∅}))
8	3, 4, 7	feq123d 5947	. . 3 ⊢ ((𝑉 UHGrph 𝐸 ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
9	2, 8	mpbid 221	. 2 ⊢ ((𝑉 UHGrph 𝐸 ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
10		eqid 2610	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
11		eqid 2610	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
12	10, 11	isuhgr 25726	. 2 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
13	9, 12	syl5ibrcom 236	1 ⊢ ((𝑉 UHGrph 𝐸 ∧ 𝑉 = (Vtx‘𝐺) ∧ 𝐸 = (iEdg‘𝐺)) → (𝐺 ∈ 𝑊 → 𝐺 ∈ UHGraph ))

Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722   UHGrph cuhg 25819

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-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

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-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  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-f 5808  df-fv 5812  df-uhgr 25724  df-uhgra 25821

This theorem is referenced by:  uhgrauhgrbi  40377