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Theorem uhgrspan1lem1 40524
 Description: Lemma 1 for uhgrspan1 40527. (Contributed by AV, 19-Nov-2020.)
Hypotheses
Ref Expression
uhgrspan1.v 𝑉 = (Vtx‘𝐺)
uhgrspan1.i 𝐼 = (iEdg‘𝐺)
uhgrspan1.f 𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}
Assertion
Ref Expression
uhgrspan1lem1 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)

Proof of Theorem uhgrspan1lem1
StepHypRef Expression
1 uhgrspan1.v . . . 4 𝑉 = (Vtx‘𝐺)
2 fvex 6113 . . . 4 (Vtx‘𝐺) ∈ V
31, 2eqeltri 2684 . . 3 𝑉 ∈ V
4 difexg 4735 . . 3 (𝑉 ∈ V → (𝑉 ∖ {𝑁}) ∈ V)
53, 4ax-mp 5 . 2 (𝑉 ∖ {𝑁}) ∈ V
6 uhgrspan1.i . . . 4 𝐼 = (iEdg‘𝐺)
7 fvex 6113 . . . 4 (iEdg‘𝐺) ∈ V
86, 7eqeltri 2684 . . 3 𝐼 ∈ V
98resex 5363 . 2 (𝐼𝐹) ∈ V
105, 9pm3.2i 470 1 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900  Vcvv 3173   ∖ cdif 3537  {csn 4125  dom cdm 5038   ↾ cres 5040  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-res 5050  df-iota 5768  df-fv 5812 This theorem is referenced by:  uhgrspan1lem2  40525  uhgrspan1lem3  40526
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