Theorem nbgraopALT 25953

Description: Alternate proof of nbgraop 25952 using mpt2xopoveq 7232, but being longer. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nbgraopALT	⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})

Distinct variable groups:   𝑛,𝑉   𝑛,𝐸   𝑛,𝑁   𝑛,𝑌   𝑛,𝑍

Proof of Theorem nbgraopALT

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nbgra 25949	. . 3 ⊢ Neighbors = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ {𝑦, 𝑛} ∈ ran (2nd ‘𝑥)})
2	1	mpt2xopoveq 7232	. 2 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd ‘𝑥)})
3		sbcel1g 3939	. . . . . 6 ⊢ (𝑁 ∈ 𝑉 → ([𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd ‘𝑥) ↔ ⦋𝑁 / 𝑦⦌{𝑦, 𝑛} ∈ ran (2nd ‘𝑥)))
4	3	adantl 481	. . . . 5 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ([𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd ‘𝑥) ↔ ⦋𝑁 / 𝑦⦌{𝑦, 𝑛} ∈ ran (2nd ‘𝑥)))
5	4	sbcbidv 3457	. . . 4 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ([⟨𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd ‘𝑥) ↔ [⟨𝑉, 𝐸⟩ / 𝑥]⦋𝑁 / 𝑦⦌{𝑦, 𝑛} ∈ ran (2nd ‘𝑥)))
6		sbcel2 3941	. . . . 5 ⊢ ([⟨𝑉, 𝐸⟩ / 𝑥]⦋𝑁 / 𝑦⦌{𝑦, 𝑛} ∈ ran (2nd ‘𝑥) ↔ ⦋𝑁 / 𝑦⦌{𝑦, 𝑛} ∈ ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌ran (2nd ‘𝑥))
7	6	a1i 11	. . . 4 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ([⟨𝑉, 𝐸⟩ / 𝑥]⦋𝑁 / 𝑦⦌{𝑦, 𝑛} ∈ ran (2nd ‘𝑥) ↔ ⦋𝑁 / 𝑦⦌{𝑦, 𝑛} ∈ ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌ran (2nd ‘𝑥)))
8		df-pr 4128	. . . . . . . 8 ⊢ {𝑦, 𝑛} = ({𝑦} ∪ {𝑛})
9	8	a1i 11	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → {𝑦, 𝑛} = ({𝑦} ∪ {𝑛}))
10	9	csbeq2dv 3944	. . . . . 6 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ⦋𝑁 / 𝑦⦌{𝑦, 𝑛} = ⦋𝑁 / 𝑦⦌({𝑦} ∪ {𝑛}))
11		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑦({𝑁} ∪ {𝑛})
12	11	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → Ⅎ𝑦({𝑁} ∪ {𝑛}))
13		sneq 4135	. . . . . . . . 9 ⊢ (𝑦 = 𝑁 → {𝑦} = {𝑁})
14	13	uneq1d 3728	. . . . . . . 8 ⊢ (𝑦 = 𝑁 → ({𝑦} ∪ {𝑛}) = ({𝑁} ∪ {𝑛}))
15	12, 14	csbiegf 3523	. . . . . . 7 ⊢ (𝑁 ∈ 𝑉 → ⦋𝑁 / 𝑦⦌({𝑦} ∪ {𝑛}) = ({𝑁} ∪ {𝑛}))
16	15	adantl 481	. . . . . 6 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ⦋𝑁 / 𝑦⦌({𝑦} ∪ {𝑛}) = ({𝑁} ∪ {𝑛}))
17		df-pr 4128	. . . . . . . 8 ⊢ {𝑁, 𝑛} = ({𝑁} ∪ {𝑛})
18	17	eqcomi 2619	. . . . . . 7 ⊢ ({𝑁} ∪ {𝑛}) = {𝑁, 𝑛}
19	18	a1i 11	. . . . . 6 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ({𝑁} ∪ {𝑛}) = {𝑁, 𝑛})
20	10, 16, 19	3eqtrd 2648	. . . . 5 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ⦋𝑁 / 𝑦⦌{𝑦, 𝑛} = {𝑁, 𝑛})
21		csbrn 5514	. . . . . . 7 ⊢ ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌ran (2nd ‘𝑥) = ran ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌(2nd ‘𝑥)
22	21	a1i 11	. . . . . 6 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌ran (2nd ‘𝑥) = ran ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌(2nd ‘𝑥))
23		opex 4859	. . . . . . . . 9 ⊢ ⟨𝑉, 𝐸⟩ ∈ V
24		csbfv2g 6142	. . . . . . . . 9 ⊢ (⟨𝑉, 𝐸⟩ ∈ V → ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌(2nd ‘𝑥) = (2nd ‘⦋⟨𝑉, 𝐸⟩ / 𝑥⦌𝑥))
25	23, 24	mp1i 13	. . . . . . . 8 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌(2nd ‘𝑥) = (2nd ‘⦋⟨𝑉, 𝐸⟩ / 𝑥⦌𝑥))
26		csbvarg 3955	. . . . . . . . . 10 ⊢ (⟨𝑉, 𝐸⟩ ∈ V → ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌𝑥 = ⟨𝑉, 𝐸⟩)
27	23, 26	mp1i 13	. . . . . . . . 9 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌𝑥 = ⟨𝑉, 𝐸⟩)
28	27	fveq2d 6107	. . . . . . . 8 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → (2nd ‘⦋⟨𝑉, 𝐸⟩ / 𝑥⦌𝑥) = (2nd ‘⟨𝑉, 𝐸⟩))
29		op2ndg 7072	. . . . . . . . 9 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
30	29	adantr 480	. . . . . . . 8 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
31	25, 28, 30	3eqtrd 2648	. . . . . . 7 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌(2nd ‘𝑥) = 𝐸)
32	31	rneqd 5274	. . . . . 6 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ran ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌(2nd ‘𝑥) = ran 𝐸)
33	22, 32	eqtrd 2644	. . . . 5 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌ran (2nd ‘𝑥) = ran 𝐸)
34	20, 33	eleq12d 2682	. . . 4 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → (⦋𝑁 / 𝑦⦌{𝑦, 𝑛} ∈ ⦋⟨𝑉, 𝐸⟩ / 𝑥⦌ran (2nd ‘𝑥) ↔ {𝑁, 𝑛} ∈ ran 𝐸))
35	5, 7, 34	3bitrd 293	. . 3 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → ([⟨𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd ‘𝑥) ↔ {𝑁, 𝑛} ∈ ran 𝐸))
36	35	rabbidv 3164	. 2 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd ‘𝑥)} = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
37	2, 36	eqtrd 2644	1 ⊢ (((𝑉 ∈ 𝑌 ∧ 𝐸 ∈ 𝑍) ∧ 𝑁 ∈ 𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  {crab 2900  Vcvv 3173  [wsbc 3402  ⦋csb 3499   ∪ cun 3538  {csn 4125  {cpr 4127  ⟨cop 4131  ran crn 5039  ‘cfv 5804  (class class class)co 6549  2^nd c2nd 7058   Neighbors cnbgra 25946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-nbgra 25949

This theorem is referenced by: (None)