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Theorem clwwlknOLD 27152
Description: Obsolete version of clwwlkn 27151 as of 22-Mar-2022. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
clwwlknOLD (𝑁 ∈ ℕ → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁})
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁

Proof of Theorem clwwlknOLD
Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-clwwlknOLD 27150 . . . . 5 ClWWalksNOLD = (𝑛 ∈ ℕ, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛})
21a1i 11 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → ClWWalksNOLD = (𝑛 ∈ ℕ, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛}))
3 fveq2 6352 . . . . . . 7 (𝑔 = 𝐺 → (ClWWalks‘𝑔) = (ClWWalks‘𝐺))
43adantl 473 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → (ClWWalks‘𝑔) = (ClWWalks‘𝐺))
5 eqeq2 2771 . . . . . . 7 (𝑛 = 𝑁 → ((♯‘𝑤) = 𝑛 ↔ (♯‘𝑤) = 𝑁))
65adantr 472 . . . . . 6 ((𝑛 = 𝑁𝑔 = 𝐺) → ((♯‘𝑤) = 𝑛 ↔ (♯‘𝑤) = 𝑁))
74, 6rabeqbidv 3335 . . . . 5 ((𝑛 = 𝑁𝑔 = 𝐺) → {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛} = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁})
87adantl 473 . . . 4 (((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) ∧ (𝑛 = 𝑁𝑔 = 𝐺)) → {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛} = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁})
9 simpl 474 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → 𝑁 ∈ ℕ)
10 simpr 479 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → 𝐺 ∈ V)
11 fvex 6362 . . . . . 6 (ClWWalks‘𝐺) ∈ V
1211rabex 4964 . . . . 5 {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ V
1312a1i 11 . . . 4 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ V)
142, 8, 9, 10, 13ovmpt2d 6953 . . 3 ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁})
1514expcom 450 . 2 (𝐺 ∈ V → (𝑁 ∈ ℕ → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁}))
161reldmmpt2 6936 . . . . 5 Rel dom ClWWalksNOLD
1716ovprc2 6848 . . . 4 𝐺 ∈ V → (𝑁ClWWalksNOLD𝐺) = ∅)
18 fvprc 6346 . . . . . 6 𝐺 ∈ V → (ClWWalks‘𝐺) = ∅)
1918rabeqdv 3334 . . . . 5 𝐺 ∈ V → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} = {𝑤 ∈ ∅ ∣ (♯‘𝑤) = 𝑁})
20 rab0 4098 . . . . 5 {𝑤 ∈ ∅ ∣ (♯‘𝑤) = 𝑁} = ∅
2119, 20syl6eq 2810 . . . 4 𝐺 ∈ V → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} = ∅)
2217, 21eqtr4d 2797 . . 3 𝐺 ∈ V → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁})
2322a1d 25 . 2 𝐺 ∈ V → (𝑁 ∈ ℕ → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁}))
2415, 23pm2.61i 176 1 (𝑁 ∈ ℕ → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  c0 4058  cfv 6049  (class class class)co 6813  cmpt2 6815  cn 11212  chash 13311  ClWWalkscclwwlk 27104  ClWWalksNOLDcclwwlknold 27148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-clwwlknOLD 27150
This theorem is referenced by:  isclwwlknOLD  27154
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