Theorem uhgrspanop 29870

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 5432	. . 3 ⊢ ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V
4	3	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V)
5	1	fvexi 6897	. . . 4 ⊢ 𝑉 ∈ V
6	2	fvexi 6897	. . . . 5 ⊢ 𝐸 ∈ V
7	6	resex 6018	. . . 4 ⊢ (𝐸 ↾ 𝐴) ∈ V
8	5, 7	opvtxfvi 29580	. . 3 ⊢ (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉
9	8	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉)
10	5, 7	opiedgfvi 29581	. . 3 ⊢ (iEdg‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = (𝐸 ↾ 𝐴)
11	10	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → (iEdg‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = (𝐸 ↾ 𝐴))
12		id 23	. 2 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
13	1, 2, 4, 9, 11, 12	uhgrspan 29866	1 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UHGraph)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ⟨cop 4590   ↾ cres 5653  ‘cfv 6537  Vtxcvtx 29567  iEdgciedg 29568  UHGraphcuhgr 29627

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-1st 7999  df-2nd 8000  df-vtx 29569  df-iedg 29570  df-edg 29619  df-uhgr 29629  df-subgr 29842

This theorem is used by: (None)