Theorem upgrspanop 29705

Description: A spanning subgraph of a pseudograph represented by an ordered pair is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 13-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
upgrspanop	⊢ (𝐺 ∈ UPGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UPGraph)

Proof of Theorem upgrspanop

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uhgrspanop.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		uhgrspanop.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3		vex 3461	. . . . . 6 ⊢ 𝑔 ∈ V
4	3	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝑔 ∈ V)
5		simprl 783	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → (Vtx‘𝑔) = 𝑉)
6		simprr 785	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → (iEdg‘𝑔) = (𝐸 ↾ 𝐴))
7		simpl 488	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝐺 ∈ UPGraph)
8	1, 2, 4, 5, 6, 7	upgrspan 29701	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝑔 ∈ UPGraph)
9	8	ex 418	. . 3 ⊢ (𝐺 ∈ UPGraph → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ UPGraph))
10	9	alrimiv 1960	. 2 ⊢ (𝐺 ∈ UPGraph → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ UPGraph))
11	1	fvexi 6899	. . 3 ⊢ 𝑉 ∈ V
12	11	a1i 11	. 2 ⊢ (𝐺 ∈ UPGraph → 𝑉 ∈ V)
13	2	fvexi 6899	. . . 4 ⊢ 𝐸 ∈ V
14	13	resex 6030	. . 3 ⊢ (𝐸 ↾ 𝐴) ∈ V
15	14	a1i 11	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 ↾ 𝐴) ∈ V)
16	10, 12, 15	gropeld 29438	1 ⊢ (𝐺 ∈ UPGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UPGraph)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597   ↾ cres 5665  ‘cfv 6540  Vtxcvtx 29401  iEdgciedg 29402  UPGraphcupgr 29485

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-1st 7992  df-2nd 7993  df-vtx 29403  df-iedg 29404  df-edg 29453  df-uhgr 29463  df-upgr 29487  df-subgr 29676

This theorem is used by: (None)