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Theorem vtxdginducedm1lem1 26643
Description: Lemma 1 for vtxdginducedm1 26647: 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 6353 . 2 (iEdg‘𝑆) = (iEdg‘⟨𝐾, 𝑃⟩)
3 vtxdginducedm1.k . . . 4 𝐾 = (𝑉 ∖ {𝑁})
4 vtxdginducedm1.v . . . . . 6 𝑉 = (Vtx‘𝐺)
54fvexi 6361 . . . . 5 𝑉 ∈ V
65difexi 4959 . . . 4 (𝑉 ∖ {𝑁}) ∈ V
73, 6eqeltri 2833 . . 3 𝐾 ∈ V
8 vtxdginducedm1.p . . . 4 𝑃 = (𝐸𝐼)
9 vtxdginducedm1.e . . . . . 6 𝐸 = (iEdg‘𝐺)
109fvexi 6361 . . . . 5 𝐸 ∈ V
1110resex 5599 . . . 4 (𝐸𝐼) ∈ V
128, 11eqeltri 2833 . . 3 𝑃 ∈ V
137, 12opiedgfvi 26087 . 2 (iEdg‘⟨𝐾, 𝑃⟩) = 𝑃
142, 13eqtri 2780 1 (iEdg‘𝑆) = 𝑃
Colors of variables: wff setvar class
Syntax hints:   = wceq 1630  wnel 3033  {crab 3052  Vcvv 3338  cdif 3710  {csn 4319  cop 4325  dom cdm 5264  cres 5266  cfv 6047  Vtxcvtx 26071  iEdgciedg 26072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-iota 6010  df-fun 6049  df-fv 6055  df-2nd 7332  df-iedg 26074
This theorem is referenced by:  vtxdginducedm1lem2  26644  vtxdginducedm1lem3  26645  vtxdginducedm1fi  26648  finsumvtxdg2ssteplem4  26652
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