Theorem vtxdginducedm1lem1 29629

Description: Lemma 1 for vtxdginducedm1 29633: 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 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
vtxdginducedm1.p	⊢ 𝑃 = (𝐸 ↾ 𝐼)
vtxdginducedm1.s	⊢ 𝑆 = ⟨𝐾, 𝑃⟩

Assertion

Ref	Expression
vtxdginducedm1lem1	⊢ (iEdg‘𝑆) = 𝑃

Proof of Theorem vtxdginducedm1lem1

Step	Hyp	Ref	Expression
1		vtxdginducedm1.s	. . 3 ⊢ 𝑆 = ⟨𝐾, 𝑃⟩
2	1	fveq2i 6845	. 2 ⊢ (iEdg‘𝑆) = (iEdg‘⟨𝐾, 𝑃⟩)
3		vtxdginducedm1.k	. . . 4 ⊢ 𝐾 = (𝑉 ∖ {𝑁})
4		vtxdginducedm1.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
5	4	fvexi 6856	. . . . 5 ⊢ 𝑉 ∈ V
6	5	difexi 5277	. . . 4 ⊢ (𝑉 ∖ {𝑁}) ∈ V
7	3, 6	eqeltri 2833	. . 3 ⊢ 𝐾 ∈ V
8		vtxdginducedm1.p	. . . 4 ⊢ 𝑃 = (𝐸 ↾ 𝐼)
9		vtxdginducedm1.e	. . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺)
10	9	fvexi 6856	. . . . 5 ⊢ 𝐸 ∈ V
11	10	resex 5996	. . . 4 ⊢ (𝐸 ↾ 𝐼) ∈ V
12	8, 11	eqeltri 2833	. . 3 ⊢ 𝑃 ∈ V
13	7, 12	opiedgfvi 29099	. 2 ⊢ (iEdg‘⟨𝐾, 𝑃⟩) = 𝑃
14	2, 13	eqtri 2760	1 ⊢ (iEdg‘𝑆) = 𝑃

Syntax hints:   = wceq 1542   ∉ wnel 3037  {crab 3401  Vcvv 3442   ∖ cdif 3900  {csn 4582  ⟨cop 4588  dom cdm 5632   ↾ cres 5634  ‘cfv 6500  Vtxcvtx 29085  iEdgciedg 29086

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fv 6508  df-2nd 7944  df-iedg 29088

This theorem is referenced by:  vtxdginducedm1lem2  29630  vtxdginducedm1lem3  29631  vtxdginducedm1fi  29634  finsumvtxdg2ssteplem4  29638