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Theorem vtxdginducedm1lem1 28487
Description: Lemma 1 for vtxdginducedm1 28491: 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 6845 . 2 (iEdg‘𝑆) = (iEdg‘⟨𝐾, 𝑃⟩)
3 vtxdginducedm1.k . . . 4 𝐾 = (𝑉 ∖ {𝑁})
4 vtxdginducedm1.v . . . . . 6 𝑉 = (Vtx‘𝐺)
54fvexi 6856 . . . . 5 𝑉 ∈ V
65difexi 5285 . . . 4 (𝑉 ∖ {𝑁}) ∈ V
73, 6eqeltri 2834 . . 3 𝐾 ∈ V
8 vtxdginducedm1.p . . . 4 𝑃 = (𝐸𝐼)
9 vtxdginducedm1.e . . . . . 6 𝐸 = (iEdg‘𝐺)
109fvexi 6856 . . . . 5 𝐸 ∈ V
1110resex 5985 . . . 4 (𝐸𝐼) ∈ V
128, 11eqeltri 2834 . . 3 𝑃 ∈ V
137, 12opiedgfvi 27961 . 2 (iEdg‘⟨𝐾, 𝑃⟩) = 𝑃
142, 13eqtri 2764 1 (iEdg‘𝑆) = 𝑃
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wnel 3049  {crab 3407  Vcvv 3445  cdif 3907  {csn 4586  cop 4592  dom cdm 5633  cres 5635  cfv 6496  Vtxcvtx 27947  iEdgciedg 27948
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-iota 6448  df-fun 6498  df-fv 6504  df-2nd 7922  df-iedg 27950
This theorem is referenced by:  vtxdginducedm1lem2  28488  vtxdginducedm1lem3  28489  vtxdginducedm1fi  28492  finsumvtxdg2ssteplem4  28496
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