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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 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {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.
StepHypRef Expression
1 df-nbgra 25949 . . 3 Neighbors = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ {𝑦, 𝑛} ∈ ran (2nd𝑥)})
21mpt2xopoveq 7232 . 2 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉[𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥)})
3 sbcel1g 3939 . . . . . 6 (𝑁𝑉 → ([𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥) ↔ 𝑁 / 𝑦{𝑦, 𝑛} ∈ ran (2nd𝑥)))
43adantl 481 . . . . 5 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ([𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥) ↔ 𝑁 / 𝑦{𝑦, 𝑛} ∈ ran (2nd𝑥)))
54sbcbidv 3457 . . . 4 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ([𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥) ↔ [𝑉, 𝐸⟩ / 𝑥]𝑁 / 𝑦{𝑦, 𝑛} ∈ ran (2nd𝑥)))
6 sbcel2 3941 . . . . 5 ([𝑉, 𝐸⟩ / 𝑥]𝑁 / 𝑦{𝑦, 𝑛} ∈ ran (2nd𝑥) ↔ 𝑁 / 𝑦{𝑦, 𝑛} ∈ 𝑉, 𝐸⟩ / 𝑥ran (2nd𝑥))
76a1i 11 . . . 4 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ([𝑉, 𝐸⟩ / 𝑥]𝑁 / 𝑦{𝑦, 𝑛} ∈ ran (2nd𝑥) ↔ 𝑁 / 𝑦{𝑦, 𝑛} ∈ 𝑉, 𝐸⟩ / 𝑥ran (2nd𝑥)))
8 df-pr 4128 . . . . . . . 8 {𝑦, 𝑛} = ({𝑦} ∪ {𝑛})
98a1i 11 . . . . . . 7 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → {𝑦, 𝑛} = ({𝑦} ∪ {𝑛}))
109csbeq2dv 3944 . . . . . 6 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑁 / 𝑦{𝑦, 𝑛} = 𝑁 / 𝑦({𝑦} ∪ {𝑛}))
11 nfcv 2751 . . . . . . . . 9 𝑦({𝑁} ∪ {𝑛})
1211a1i 11 . . . . . . . 8 (𝑁𝑉𝑦({𝑁} ∪ {𝑛}))
13 sneq 4135 . . . . . . . . 9 (𝑦 = 𝑁 → {𝑦} = {𝑁})
1413uneq1d 3728 . . . . . . . 8 (𝑦 = 𝑁 → ({𝑦} ∪ {𝑛}) = ({𝑁} ∪ {𝑛}))
1512, 14csbiegf 3523 . . . . . . 7 (𝑁𝑉𝑁 / 𝑦({𝑦} ∪ {𝑛}) = ({𝑁} ∪ {𝑛}))
1615adantl 481 . . . . . 6 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑁 / 𝑦({𝑦} ∪ {𝑛}) = ({𝑁} ∪ {𝑛}))
17 df-pr 4128 . . . . . . . 8 {𝑁, 𝑛} = ({𝑁} ∪ {𝑛})
1817eqcomi 2619 . . . . . . 7 ({𝑁} ∪ {𝑛}) = {𝑁, 𝑛}
1918a1i 11 . . . . . 6 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ({𝑁} ∪ {𝑛}) = {𝑁, 𝑛})
2010, 16, 193eqtrd 2648 . . . . 5 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑁 / 𝑦{𝑦, 𝑛} = {𝑁, 𝑛})
21 csbrn 5514 . . . . . . 7 𝑉, 𝐸⟩ / 𝑥ran (2nd𝑥) = ran 𝑉, 𝐸⟩ / 𝑥(2nd𝑥)
2221a1i 11 . . . . . 6 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑉, 𝐸⟩ / 𝑥ran (2nd𝑥) = ran 𝑉, 𝐸⟩ / 𝑥(2nd𝑥))
23 opex 4859 . . . . . . . . 9 𝑉, 𝐸⟩ ∈ V
24 csbfv2g 6142 . . . . . . . . 9 (⟨𝑉, 𝐸⟩ ∈ V → 𝑉, 𝐸⟩ / 𝑥(2nd𝑥) = (2nd𝑉, 𝐸⟩ / 𝑥𝑥))
2523, 24mp1i 13 . . . . . . . 8 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑉, 𝐸⟩ / 𝑥(2nd𝑥) = (2nd𝑉, 𝐸⟩ / 𝑥𝑥))
26 csbvarg 3955 . . . . . . . . . 10 (⟨𝑉, 𝐸⟩ ∈ V → 𝑉, 𝐸⟩ / 𝑥𝑥 = ⟨𝑉, 𝐸⟩)
2723, 26mp1i 13 . . . . . . . . 9 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑉, 𝐸⟩ / 𝑥𝑥 = ⟨𝑉, 𝐸⟩)
2827fveq2d 6107 . . . . . . . 8 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (2nd𝑉, 𝐸⟩ / 𝑥𝑥) = (2nd ‘⟨𝑉, 𝐸⟩))
29 op2ndg 7072 . . . . . . . . 9 ((𝑉𝑌𝐸𝑍) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
3029adantr 480 . . . . . . . 8 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
3125, 28, 303eqtrd 2648 . . . . . . 7 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑉, 𝐸⟩ / 𝑥(2nd𝑥) = 𝐸)
3231rneqd 5274 . . . . . 6 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ran 𝑉, 𝐸⟩ / 𝑥(2nd𝑥) = ran 𝐸)
3322, 32eqtrd 2644 . . . . 5 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑉, 𝐸⟩ / 𝑥ran (2nd𝑥) = ran 𝐸)
3420, 33eleq12d 2682 . . . 4 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (𝑁 / 𝑦{𝑦, 𝑛} ∈ 𝑉, 𝐸⟩ / 𝑥ran (2nd𝑥) ↔ {𝑁, 𝑛} ∈ ran 𝐸))
355, 7, 343bitrd 293 . . 3 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ([𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥) ↔ {𝑁, 𝑛} ∈ ran 𝐸))
3635rabbidv 3164 . 2 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → {𝑛𝑉[𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥)} = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
372, 36eqtrd 2644 1 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
 Colors of variables: wff setvar class 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  2nd 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)
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