Theorem upgrspanop 29314

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 3484	. . . . . 6 ⊢ 𝑔 ∈ V
4	3	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝑔 ∈ V)
5		simprl 771	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → (Vtx‘𝑔) = 𝑉)
6		simprr 773	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → (iEdg‘𝑔) = (𝐸 ↾ 𝐴))
7		simpl 482	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝐺 ∈ UPGraph)
8	1, 2, 4, 5, 6, 7	upgrspan 29310	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝑔 ∈ UPGraph)
9	8	ex 412	. . 3 ⊢ (𝐺 ∈ UPGraph → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ UPGraph))
10	9	alrimiv 1927	. 2 ⊢ (𝐺 ∈ UPGraph → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ UPGraph))
11	1	fvexi 6920	. . 3 ⊢ 𝑉 ∈ V
12	11	a1i 11	. 2 ⊢ (𝐺 ∈ UPGraph → 𝑉 ∈ V)
13	2	fvexi 6920	. . . 4 ⊢ 𝐸 ∈ V
14	13	resex 6047	. . 3 ⊢ (𝐸 ↾ 𝐴) ∈ V
15	14	a1i 11	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 ↾ 𝐴) ∈ V)
16	10, 12, 15	gropeld 29050	1 ⊢ (𝐺 ∈ UPGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UPGraph)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ⟨cop 4632   ↾ cres 5687  ‘cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  UPGraphcupgr 29097

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-1st 8014  df-2nd 8015  df-vtx 29015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-subgr 29285

This theorem is referenced by: (None)