Theorem uhgrspanop 29379

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 5411	. . 3 ⊢ ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V
4	3	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V)
5	1	fvexi 6848	. . . 4 ⊢ 𝑉 ∈ V
6	2	fvexi 6848	. . . . 5 ⊢ 𝐸 ∈ V
7	6	resex 5988	. . . 4 ⊢ (𝐸 ↾ 𝐴) ∈ V
8	5, 7	opvtxfvi 29092	. . 3 ⊢ (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉
9	8	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉)
10	5, 7	opiedgfvi 29093	. . 3 ⊢ (iEdg‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = (𝐸 ↾ 𝐴)
11	10	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → (iEdg‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = (𝐸 ↾ 𝐴))
12		id 22	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
13	1, 2, 4, 9, 11, 12	uhgrspan 29375	1 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UHGraph)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574   ↾ cres 5626  ‘cfv 6492  Vtxcvtx 29079  iEdgciedg 29080  UHGraphcuhgr 29139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-1st 7935  df-2nd 7936  df-vtx 29081  df-iedg 29082  df-edg 29131  df-uhgr 29141  df-subgr 29351

This theorem is referenced by: (None)