Theorem vtxdginducedm1lem1 29575

Description: Lemma 1 for vtxdginducedm1 29579: 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 6923	. 2 ⊢ (iEdg‘𝑆) = (iEdg‘⟨𝐾, 𝑃⟩)
3		vtxdginducedm1.k	. . . 4 ⊢ 𝐾 = (𝑉 ∖ {𝑁})
4		vtxdginducedm1.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
5	4	fvexi 6934	. . . . 5 ⊢ 𝑉 ∈ V
6	5	difexi 5348	. . . 4 ⊢ (𝑉 ∖ {𝑁}) ∈ V
7	3, 6	eqeltri 2840	. . 3 ⊢ 𝐾 ∈ V
8		vtxdginducedm1.p	. . . 4 ⊢ 𝑃 = (𝐸 ↾ 𝐼)
9		vtxdginducedm1.e	. . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺)
10	9	fvexi 6934	. . . . 5 ⊢ 𝐸 ∈ V
11	10	resex 6058	. . . 4 ⊢ (𝐸 ↾ 𝐼) ∈ V
12	8, 11	eqeltri 2840	. . 3 ⊢ 𝑃 ∈ V
13	7, 12	opiedgfvi 29045	. 2 ⊢ (iEdg‘⟨𝐾, 𝑃⟩) = 𝑃
14	2, 13	eqtri 2768	1 ⊢ (iEdg‘𝑆) = 𝑃

Syntax hints:   = wceq 1537   ∉ wnel 3052  {crab 3443  Vcvv 3488   ∖ cdif 3973  {csn 4648  ⟨cop 4654  dom cdm 5700   ↾ cres 5702  ‘cfv 6573  Vtxcvtx 29031  iEdgciedg 29032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581  df-2nd 8031  df-iedg 29034

This theorem is referenced by:  vtxdginducedm1lem2  29576  vtxdginducedm1lem3  29577  vtxdginducedm1fi  29580  finsumvtxdg2ssteplem4  29584