Theorem vtxdginducedm1lem1 29608

Description: Lemma 1 for vtxdginducedm1 29612: 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 6843	. 2 ⊢ (iEdg‘𝑆) = (iEdg‘⟨𝐾, 𝑃⟩)
3		vtxdginducedm1.k	. . . 4 ⊢ 𝐾 = (𝑉 ∖ {𝑁})
4		vtxdginducedm1.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
5	4	fvexi 6854	. . . . 5 ⊢ 𝑉 ∈ V
6	5	difexi 5271	. . . 4 ⊢ (𝑉 ∖ {𝑁}) ∈ V
7	3, 6	eqeltri 2832	. . 3 ⊢ 𝐾 ∈ V
8		vtxdginducedm1.p	. . . 4 ⊢ 𝑃 = (𝐸 ↾ 𝐼)
9		vtxdginducedm1.e	. . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺)
10	9	fvexi 6854	. . . . 5 ⊢ 𝐸 ∈ V
11	10	resex 5994	. . . 4 ⊢ (𝐸 ↾ 𝐼) ∈ V
12	8, 11	eqeltri 2832	. . 3 ⊢ 𝑃 ∈ V
13	7, 12	opiedgfvi 29079	. 2 ⊢ (iEdg‘⟨𝐾, 𝑃⟩) = 𝑃
14	2, 13	eqtri 2759	1 ⊢ (iEdg‘𝑆) = 𝑃

Syntax hints:   = wceq 1542   ∉ wnel 3036  {crab 3389  Vcvv 3429   ∖ cdif 3886  {csn 4567  ⟨cop 4573  dom cdm 5631   ↾ cres 5633  ‘cfv 6498  Vtxcvtx 29065  iEdgciedg 29066

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 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6454  df-fun 6500  df-fv 6506  df-2nd 7943  df-iedg 29068

This theorem is referenced by:  vtxdginducedm1lem2  29609  vtxdginducedm1lem3  29610  vtxdginducedm1fi  29613  finsumvtxdg2ssteplem4  29617