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Theorem uhgrspan1lem2 26086
Description: Lemma 2 for uhgrspan1 26088. (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)}
uhgrspan1.s 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩
Assertion
Ref Expression
uhgrspan1lem2 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})

Proof of Theorem uhgrspan1lem2
StepHypRef Expression
1 uhgrspan1.s . . 3 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩
21fveq2i 6151 . 2 (Vtx‘𝑆) = (Vtx‘⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩)
3 uhgrspan1.v . . . 4 𝑉 = (Vtx‘𝐺)
4 uhgrspan1.i . . . 4 𝐼 = (iEdg‘𝐺)
5 uhgrspan1.f . . . 4 𝐹 = {𝑖 ∈ dom 𝐼𝑁 ∉ (𝐼𝑖)}
63, 4, 5uhgrspan1lem1 26085 . . 3 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)
7 opvtxfv 25784 . . 3 (((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V) → (Vtx‘⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩) = (𝑉 ∖ {𝑁}))
86, 7ax-mp 5 . 2 (Vtx‘⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩) = (𝑉 ∖ {𝑁})
92, 8eqtri 2643 1 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wcel 1987  wnel 2893  {crab 2911  Vcvv 3186  cdif 3552  {csn 4148  cop 4154  dom cdm 5074  cres 5076  cfv 5847  Vtxcvtx 25774  iEdgciedg 25775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fv 5855  df-1st 7113  df-vtx 25776
This theorem is referenced by:  uhgrspan1  26088
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