Theorem uhgrspanop 26233

Description: A spanning subgraph of a hypergraph represented by an ordered pair is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.)

Hypotheses

Ref	Expression
uhgrspanop.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgrspanop.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
uhgrspanop	⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UHGraph)

Proof of Theorem uhgrspanop

Step	Hyp	Ref	Expression
1		uhgrspanop.v	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
2		uhgrspanop.e	. 2 ⊢ 𝐸 = (iEdg‘𝐺)
3		opex 4962	. . 3 ⊢ ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V
4	3	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V)
5		fvex 6239	. . . . 5 ⊢ (Vtx‘𝐺) ∈ V
6	1, 5	eqeltri 2726	. . . 4 ⊢ 𝑉 ∈ V
7		fvex 6239	. . . . . 6 ⊢ (iEdg‘𝐺) ∈ V
8	2, 7	eqeltri 2726	. . . . 5 ⊢ 𝐸 ∈ V
9	8	resex 5478	. . . 4 ⊢ (𝐸 ↾ 𝐴) ∈ V
10		opvtxfv 25929	. . . 4 ⊢ ((𝑉 ∈ V ∧ (𝐸 ↾ 𝐴) ∈ V) → (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉)
11	6, 9, 10	mp2an 708	. . 3 ⊢ (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉
12	11	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉)
13		opiedgfv 25932	. . . 4 ⊢ ((𝑉 ∈ V ∧ (𝐸 ↾ 𝐴) ∈ V) → (iEdg‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = (𝐸 ↾ 𝐴))
14	6, 9, 13	mp2an 708	. . 3 ⊢ (iEdg‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = (𝐸 ↾ 𝐴)
15	14	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → (iEdg‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = (𝐸 ↾ 𝐴))
16		id 22	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
17	1, 2, 4, 12, 15, 16	uhgrspan 26229	1 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UHGraph)

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ⟨cop 4216   ↾ cres 5145  ‘cfv 5926  Vtxcvtx 25919  iEdgciedg 25920  UHGraphcuhgr 25996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-1st 7210  df-2nd 7211  df-vtx 25921  df-iedg 25922  df-edg 25985  df-uhgr 25998  df-subgr 26205

This theorem is referenced by: (None)