Theorem vtxdginducedm1lem3 27326

Description: Lemma 3 for vtxdginducedm1 27328: an edge 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
vtxdginducedm1lem3	⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻))

Distinct variable group:   𝑖,𝐸

Allowed substitution hints:   𝑃(𝑖)   𝑆(𝑖)   𝐺(𝑖)   𝐻(𝑖)   𝐼(𝑖)   𝐾(𝑖)   𝑁(𝑖)   𝑉(𝑖)

Proof of Theorem vtxdginducedm1lem3

Step	Hyp	Ref	Expression
1		vtxdginducedm1.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		vtxdginducedm1.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3		vtxdginducedm1.k	. . . . 5 ⊢ 𝐾 = (𝑉 ∖ {𝑁})
4		vtxdginducedm1.i	. . . . 5 ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}
5		vtxdginducedm1.p	. . . . 5 ⊢ 𝑃 = (𝐸 ↾ 𝐼)
6		vtxdginducedm1.s	. . . . 5 ⊢ 𝑆 = ⟨𝐾, 𝑃⟩
7	1, 2, 3, 4, 5, 6	vtxdginducedm1lem1 27324	. . . 4 ⊢ (iEdg‘𝑆) = 𝑃
8	7, 5	eqtri 2847	. . 3 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐼)
9	8	fveq1i 6674	. 2 ⊢ ((iEdg‘𝑆)‘𝐻) = ((𝐸 ↾ 𝐼)‘𝐻)
10		fvres 6692	. 2 ⊢ (𝐻 ∈ 𝐼 → ((𝐸 ↾ 𝐼)‘𝐻) = (𝐸‘𝐻))
11	9, 10	syl5eq 2871	1 ⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113   ∉ wnel 3126  {crab 3145   ∖ cdif 3936  {csn 4570  ⟨cop 4576  dom cdm 5558   ↾ cres 5560  ‘cfv 6358  Vtxcvtx 26784  iEdgciedg 26785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-iota 6317  df-fun 6360  df-fv 6366  df-2nd 7693  df-iedg 26787

This theorem is referenced by:  vtxdginducedm1  27328