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Theorem nbgraopALT 25719
Description: Alternate proof of nbgraop 25718 using mpt2xopoveq 7209, 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 25715 . . 3 Neighbors = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ {𝑦, 𝑛} ∈ ran (2nd𝑥)})
21mpt2xopoveq 7209 . 2 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉[𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥)})
3 sbcel1g 3938 . . . . . 6 (𝑁𝑉 → ([𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥) ↔ 𝑁 / 𝑦{𝑦, 𝑛} ∈ ran (2nd𝑥)))
43adantl 480 . . . . 5 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ([𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥) ↔ 𝑁 / 𝑦{𝑦, 𝑛} ∈ ran (2nd𝑥)))
54sbcbidv 3456 . . . 4 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ([𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥) ↔ [𝑉, 𝐸⟩ / 𝑥]𝑁 / 𝑦{𝑦, 𝑛} ∈ ran (2nd𝑥)))
6 sbcel2 3940 . . . . 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 4127 . . . . . . . 8 {𝑦, 𝑛} = ({𝑦} ∪ {𝑛})
98a1i 11 . . . . . . 7 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → {𝑦, 𝑛} = ({𝑦} ∪ {𝑛}))
109csbeq2dv 3943 . . . . . 6 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑁 / 𝑦{𝑦, 𝑛} = 𝑁 / 𝑦({𝑦} ∪ {𝑛}))
11 nfcv 2750 . . . . . . . . 9 𝑦({𝑁} ∪ {𝑛})
1211a1i 11 . . . . . . . 8 (𝑁𝑉𝑦({𝑁} ∪ {𝑛}))
13 sneq 4134 . . . . . . . . 9 (𝑦 = 𝑁 → {𝑦} = {𝑁})
1413uneq1d 3727 . . . . . . . 8 (𝑦 = 𝑁 → ({𝑦} ∪ {𝑛}) = ({𝑁} ∪ {𝑛}))
1512, 14csbiegf 3522 . . . . . . 7 (𝑁𝑉𝑁 / 𝑦({𝑦} ∪ {𝑛}) = ({𝑁} ∪ {𝑛}))
1615adantl 480 . . . . . 6 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑁 / 𝑦({𝑦} ∪ {𝑛}) = ({𝑁} ∪ {𝑛}))
17 df-pr 4127 . . . . . . . 8 {𝑁, 𝑛} = ({𝑁} ∪ {𝑛})
1817eqcomi 2618 . . . . . . 7 ({𝑁} ∪ {𝑛}) = {𝑁, 𝑛}
1918a1i 11 . . . . . 6 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ({𝑁} ∪ {𝑛}) = {𝑁, 𝑛})
2010, 16, 193eqtrd 2647 . . . . 5 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑁 / 𝑦{𝑦, 𝑛} = {𝑁, 𝑛})
21 csbrn 5500 . . . . . . 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 4853 . . . . . . . . 9 𝑉, 𝐸⟩ ∈ V
24 csbfv2g 6127 . . . . . . . . 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 3954 . . . . . . . . . 10 (⟨𝑉, 𝐸⟩ ∈ V → 𝑉, 𝐸⟩ / 𝑥𝑥 = ⟨𝑉, 𝐸⟩)
2723, 26mp1i 13 . . . . . . . . 9 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑉, 𝐸⟩ / 𝑥𝑥 = ⟨𝑉, 𝐸⟩)
2827fveq2d 6092 . . . . . . . 8 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (2nd𝑉, 𝐸⟩ / 𝑥𝑥) = (2nd ‘⟨𝑉, 𝐸⟩))
29 op2ndg 7049 . . . . . . . . 9 ((𝑉𝑌𝐸𝑍) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
3029adantr 479 . . . . . . . 8 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
3125, 28, 303eqtrd 2647 . . . . . . 7 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑉, 𝐸⟩ / 𝑥(2nd𝑥) = 𝐸)
3231rneqd 5261 . . . . . 6 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ran 𝑉, 𝐸⟩ / 𝑥(2nd𝑥) = ran 𝐸)
3322, 32eqtrd 2643 . . . . 5 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → 𝑉, 𝐸⟩ / 𝑥ran (2nd𝑥) = ran 𝐸)
3420, 33eleq12d 2681 . . . 4 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (𝑁 / 𝑦{𝑦, 𝑛} ∈ 𝑉, 𝐸⟩ / 𝑥ran (2nd𝑥) ↔ {𝑁, 𝑛} ∈ ran 𝐸))
355, 7, 343bitrd 292 . . 3 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → ([𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥) ↔ {𝑁, 𝑛} ∈ ran 𝐸))
3635rabbidv 3163 . 2 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → {𝑛𝑉[𝑉, 𝐸⟩ / 𝑥][𝑁 / 𝑦]{𝑦, 𝑛} ∈ ran (2nd𝑥)} = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
372, 36eqtrd 2643 1 (((𝑉𝑌𝐸𝑍) ∧ 𝑁𝑉) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = {𝑛𝑉 ∣ {𝑁, 𝑛} ∈ ran 𝐸})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wnfc 2737  {crab 2899  Vcvv 3172  [wsbc 3401  csb 3498  cun 3537  {csn 4124  {cpr 4126  cop 4130  ran crn 5029  cfv 5790  (class class class)co 6527  2nd c2nd 7035   Neighbors cnbgra 25712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-nbgra 25715
This theorem is referenced by: (None)
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