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Theorem upgrspanop 27186
 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, (𝐸𝐴)⟩ ∈ UPGraph)

Proof of Theorem upgrspanop
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 uhgrspanop.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 uhgrspanop.e . . . . 5 𝐸 = (iEdg‘𝐺)
3 vex 3413 . . . . . 6 𝑔 ∈ V
43a1i 11 . . . . 5 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → 𝑔 ∈ V)
5 simprl 770 . . . . 5 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → (Vtx‘𝑔) = 𝑉)
6 simprr 772 . . . . 5 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → (iEdg‘𝑔) = (𝐸𝐴))
7 simpl 486 . . . . 5 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → 𝐺 ∈ UPGraph)
81, 2, 4, 5, 6, 7upgrspan 27182 . . . 4 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → 𝑔 ∈ UPGraph)
98ex 416 . . 3 (𝐺 ∈ UPGraph → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴)) → 𝑔 ∈ UPGraph))
109alrimiv 1928 . 2 (𝐺 ∈ UPGraph → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴)) → 𝑔 ∈ UPGraph))
111fvexi 6672 . . 3 𝑉 ∈ V
1211a1i 11 . 2 (𝐺 ∈ UPGraph → 𝑉 ∈ V)
132fvexi 6672 . . . 4 𝐸 ∈ V
1413resex 5871 . . 3 (𝐸𝐴) ∈ V
1514a1i 11 . 2 (𝐺 ∈ UPGraph → (𝐸𝐴) ∈ V)
1610, 12, 15gropeld 26925 1 (𝐺 ∈ UPGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UPGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4528   ↾ cres 5526  ‘cfv 6335  Vtxcvtx 26888  iEdgciedg 26889  UPGraphcupgr 26972 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-1st 7693  df-2nd 7694  df-vtx 26890  df-iedg 26891  df-edg 26940  df-uhgr 26950  df-upgr 26974  df-subgr 27157 This theorem is referenced by: (None)
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