Theorem vtxdginducedm1lem1 30002

Description: Lemma 1 for vtxdginducedm1 30006: 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 6882	. 2 ⊢ (iEdg‘𝑆) = (iEdg‘⟨𝐾, 𝑃⟩)
3		vtxdginducedm1.k	. . . 4 ⊢ 𝐾 = (𝑉 ∖ {𝑁})
4		vtxdginducedm1.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
5	4	fvexi 6893	. . . . 5 ⊢ 𝑉 ∈ V
6	5	difexi 5295	. . . 4 ⊢ (𝑉 ∖ {𝑁}) ∈ V
7	3, 6	eqeltri 2856	. . 3 ⊢ 𝐾 ∈ V
8		vtxdginducedm1.p	. . . 4 ⊢ 𝑃 = (𝐸 ↾ 𝐼)
9		vtxdginducedm1.e	. . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺)
10	9	fvexi 6893	. . . . 5 ⊢ 𝐸 ∈ V
11	10	resex 6022	. . . 4 ⊢ (𝐸 ↾ 𝐼) ∈ V
12	8, 11	eqeltri 2856	. . 3 ⊢ 𝑃 ∈ V
13	7, 12	opiedgfvi 29470	. 2 ⊢ (iEdg‘⟨𝐾, 𝑃⟩) = 𝑃
14	2, 13	eqtri 2783	1 ⊢ (iEdg‘𝑆) = 𝑃

This proof depends on syntax axioms:   = wceq 1570   ∉ wnel 3061  {crab 3412  Vcvv 3450   ∖ cdif 3896  {csn 4584  ⟨cop 4590  dom cdm 5655   ↾ cres 5657  ‘cfv 6533  Vtxcvtx 29456  iEdgciedg 29457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-iota 6489  df-fun 6535  df-fv 6541  df-2nd 7988  df-iedg 29459

This theorem is used by:  vtxdginducedm1lem2  30003  vtxdginducedm1lem3  30004  vtxdginducedm1fi  30007  finsumvtxdg2ssteplem4  30011