Theorem uhgrspanop 29259

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 6840	. . . 4 ⊢ 𝑉 ∈ V
6	2	fvexi 6840	. . . . 5 ⊢ 𝐸 ∈ V
7	6	resex 5984	. . . 4 ⊢ (𝐸 ↾ 𝐴) ∈ V
8	5, 7	opvtxfvi 28972	. . 3 ⊢ (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉
9	8	a1i 11	. 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉)
10	5, 7	opiedgfvi 28973	. . 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 29255	1 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UHGraph)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ⟨cop 4585   ↾ cres 5625  ‘cfv 6486  Vtxcvtx 28959  iEdgciedg 28960  UHGraphcuhgr 29019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-1st 7931  df-2nd 7932  df-vtx 28961  df-iedg 28962  df-edg 29011  df-uhgr 29021  df-subgr 29231

This theorem is referenced by: (None)