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Theorem uhgrspan1lem3 27081
Description: Lemma 3 for uhgrspan1 27082. (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
uhgrspan1lem3 (iEdg‘𝑆) = (𝐼𝐹)

Proof of Theorem uhgrspan1lem3
StepHypRef Expression
1 uhgrspan1.s . . 3 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩
21fveq2i 6654 . 2 (iEdg‘𝑆) = (iEdg‘⟨(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 27079 . . 3 ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V)
7 opiedgfv 26789 . . 3 (((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼𝐹) ∈ V) → (iEdg‘⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩) = (𝐼𝐹))
86, 7ax-mp 5 . 2 (iEdg‘⟨(𝑉 ∖ {𝑁}), (𝐼𝐹)⟩) = (𝐼𝐹)
92, 8eqtri 2847 1 (iEdg‘𝑆) = (𝐼𝐹)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2115  wnel 3117  {crab 3136  Vcvv 3479  cdif 3915  {csn 4548  cop 4554  dom cdm 5536  cres 5538  cfv 6336  Vtxcvtx 26778  iEdgciedg 26779
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-iota 6295  df-fun 6338  df-fv 6344  df-2nd 7673  df-iedg 26781
This theorem is referenced by:  uhgrspan1  27082  upgrres  27085  umgrres  27086  usgrres  27087
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