Theorem uhgrspanop 29923

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 5437	. . 3 ⊢ ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V
4	3	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V)
5	1	fvexi 6928	. . . 4 ⊢ 𝑉 ∈ V
6	2	fvexi 6928	. . . . 5 ⊢ 𝐸 ∈ V
7	6	resex 6034	. . . 4 ⊢ (𝐸 ↾ 𝐴) ∈ V
8	5, 7	opvtxfvi 29633	. . 3 ⊢ (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉
9	8	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉)
10	5, 7	opiedgfvi 29634	. . 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 29919	1 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UHGraph)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4838   ↾ cres 5658  ‘cfv 6564  Vtxcvtx 29620  iEdgciedg 29621  UHGraphcuhgr 29680

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7781

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-fv 6572  df-1st 8033  df-2nd 8034  df-vtx 29622  df-iedg 29623  df-edg 29672  df-uhgr 29682  df-subgr 29895

This theorem is used by: (None)