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.

Step	Hyp	Ref	Expression
1		df-clwwlknOLD 27150	. . . . 5 ⊢ ClWWalksNOLD = (𝑛 ∈ ℕ, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛})
2	1	a1i 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))
4	3	adantl 473	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (ClWWalks‘𝑔) = (ClWWalks‘𝐺))
5		eqeq2 2771	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((♯‘𝑤) = 𝑛 ↔ (♯‘𝑤) = 𝑁))
6	5	adantr 472	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((♯‘𝑤) = 𝑛 ↔ (♯‘𝑤) = 𝑁))
7	4, 6	rabeqbidv 3335	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛} = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁})
8	7	adantl 473	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) ∧ (𝑛 = 𝑁 ∧ 𝑔 = 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
12	11	rabex 4964	. . . . 5 ⊢ {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ V
13	12	a1i 11	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ V)
14	2, 8, 9, 10, 13	ovmpt2d 6953	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ V) → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁})
15	14	expcom 450	. 2 ⊢ (𝐺 ∈ V → (𝑁 ∈ ℕ → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁}))
16	1	reldmmpt2 6936	. . . . 5 ⊢ Rel dom ClWWalksNOLD
17	16	ovprc2 6848	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁ClWWalksNOLD𝐺) = ∅)
18		fvprc 6346	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (ClWWalks‘𝐺) = ∅)
19	18	rabeqdv 3334	. . . . 5 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} = {𝑤 ∈ ∅ ∣ (♯‘𝑤) = 𝑁})
20		rab0 4098	. . . . 5 ⊢ {𝑤 ∈ ∅ ∣ (♯‘𝑤) = 𝑁} = ∅
21	19, 20	syl6eq 2810	. . . 4 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} = ∅)
22	17, 21	eqtr4d 2797	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁})
23	22	a1d 25	. 2 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∈ ℕ → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁}))
24	15, 23	pm2.61i 176	1 ⊢ (𝑁 ∈ ℕ → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁})

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  ℕcn 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