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Theorem upgrspanop 28294
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 3451 . . . . . 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 484 . . . . 5 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → 𝐺 ∈ UPGraph)
81, 2, 4, 5, 6, 7upgrspan 28290 . . . 4 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → 𝑔 ∈ UPGraph)
98ex 414 . . 3 (𝐺 ∈ UPGraph → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴)) → 𝑔 ∈ UPGraph))
109alrimiv 1931 . 2 (𝐺 ∈ UPGraph → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴)) → 𝑔 ∈ UPGraph))
111fvexi 6860 . . 3 𝑉 ∈ V
1211a1i 11 . 2 (𝐺 ∈ UPGraph → 𝑉 ∈ V)
132fvexi 6860 . . . 4 𝐸 ∈ V
1413resex 5989 . . 3 (𝐸𝐴) ∈ V
1514a1i 11 . 2 (𝐺 ∈ UPGraph → (𝐸𝐴) ∈ V)
1610, 12, 15gropeld 28033 1 (𝐺 ∈ UPGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3447  cop 4596  cres 5639  cfv 6500  Vtxcvtx 27996  iEdgciedg 27997  UPGraphcupgr 28080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-1st 7925  df-2nd 7926  df-vtx 27998  df-iedg 27999  df-edg 28048  df-uhgr 28058  df-upgr 28082  df-subgr 28265
This theorem is referenced by: (None)
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