Theorem upgrspanop 29270

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 3440	. . . . . 6 ⊢ 𝑔 ∈ V
4	3	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → 𝑔 ∈ V)
5		simprl 770	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴))) → (Vtx‘𝑔) = 𝑉)
6		simprr 772	. . . . 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 29266	. . . 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 1928	. 2 ⊢ (𝐺 ∈ UPGraph → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸 ↾ 𝐴)) → 𝑔 ∈ UPGraph))
11	1	fvexi 6831	. . 3 ⊢ 𝑉 ∈ V
12	11	a1i 11	. 2 ⊢ (𝐺 ∈ UPGraph → 𝑉 ∈ V)
13	2	fvexi 6831	. . . 4 ⊢ 𝐸 ∈ V
14	13	resex 5973	. . 3 ⊢ (𝐸 ↾ 𝐴) ∈ V
15	14	a1i 11	. 2 ⊢ (𝐺 ∈ UPGraph → (𝐸 ↾ 𝐴) ∈ V)
16	10, 12, 15	gropeld 29006	1 ⊢ (𝐺 ∈ UPGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UPGraph)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4577   ↾ cres 5613  ‘cfv 6476  Vtxcvtx 28969  iEdgciedg 28970  UPGraphcupgr 29053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-1st 7916  df-2nd 7917  df-vtx 28971  df-iedg 28972  df-edg 29021  df-uhgr 29031  df-upgr 29055  df-subgr 29241

This theorem is referenced by: (None)