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Theorem vtxdginducedm1lem1 27373
 Description: Lemma 1 for vtxdginducedm1 27377: the edge function in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.)
Hypotheses
Ref Expression
vtxdginducedm1.v 𝑉 = (Vtx‘𝐺)
vtxdginducedm1.e 𝐸 = (iEdg‘𝐺)
vtxdginducedm1.k 𝐾 = (𝑉 ∖ {𝑁})
vtxdginducedm1.i 𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}
vtxdginducedm1.p 𝑃 = (𝐸𝐼)
vtxdginducedm1.s 𝑆 = ⟨𝐾, 𝑃
Assertion
Ref Expression
vtxdginducedm1lem1 (iEdg‘𝑆) = 𝑃

Proof of Theorem vtxdginducedm1lem1
StepHypRef Expression
1 vtxdginducedm1.s . . 3 𝑆 = ⟨𝐾, 𝑃
21fveq2i 6658 . 2 (iEdg‘𝑆) = (iEdg‘⟨𝐾, 𝑃⟩)
3 vtxdginducedm1.k . . . 4 𝐾 = (𝑉 ∖ {𝑁})
4 vtxdginducedm1.v . . . . . 6 𝑉 = (Vtx‘𝐺)
54fvexi 6669 . . . . 5 𝑉 ∈ V
65difexi 5200 . . . 4 (𝑉 ∖ {𝑁}) ∈ V
73, 6eqeltri 2886 . . 3 𝐾 ∈ V
8 vtxdginducedm1.p . . . 4 𝑃 = (𝐸𝐼)
9 vtxdginducedm1.e . . . . . 6 𝐸 = (iEdg‘𝐺)
109fvexi 6669 . . . . 5 𝐸 ∈ V
1110resex 5870 . . . 4 (𝐸𝐼) ∈ V
128, 11eqeltri 2886 . . 3 𝑃 ∈ V
137, 12opiedgfvi 26847 . 2 (iEdg‘⟨𝐾, 𝑃⟩) = 𝑃
142, 13eqtri 2821 1 (iEdg‘𝑆) = 𝑃
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∉ wnel 3091  {crab 3110  Vcvv 3442   ∖ cdif 3880  {csn 4528  ⟨cop 4534  dom cdm 5523   ↾ cres 5525  ‘cfv 6332  Vtxcvtx 26833  iEdgciedg 26834 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 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6291  df-fun 6334  df-fv 6340  df-2nd 7685  df-iedg 26836 This theorem is referenced by:  vtxdginducedm1lem2  27374  vtxdginducedm1lem3  27375  vtxdginducedm1fi  27378  finsumvtxdg2ssteplem4  27382
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