Theorem cntzex 14144

Description: Set existence of the centralizer. (Contributed by Jim Kingdon, 15-Sep-2026.)

Hypothesis

Ref	Expression
cntzex.z	⊢ 𝑍 = (Cntz‘𝑀)

Assertion

Ref	Expression
cntzex	⊢ (𝑀 ∈ 𝑉 → 𝑍 ∈ V)

Proof of Theorem cntzex

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

Step	Hyp	Ref	Expression
1		cntzex.z	. 2 ⊢ 𝑍 = (Cntz‘𝑀)
2		df-cntz 14142	. . . 4 ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
3		fveq2 5695	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4	3	pweqd 3693	. . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
5		fveq2 5695	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
6	5	oveqd 6102	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥(+g‘𝑀)𝑦))
7	5	oveqd 6102	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑦(+g‘𝑚)𝑥) = (𝑦(+g‘𝑀)𝑥))
8	6, 7	eqeq12d 2253	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))
9	8	ralbidv 2550	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)))
10	3, 9	rabeqbidv 2816	. . . . 5 ⊢ (𝑚 = 𝑀 → {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)} = {𝑥 ∈ (Base‘𝑀) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)})
11	4, 10	mpteq12dv 4213	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}) = (𝑠 ∈ 𝒫 (Base‘𝑀) ↦ {𝑥 ∈ (Base‘𝑀) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}))
12		elex 2833	. . . 4 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
13		basfn 13463	. . . . . . 7 ⊢ Base Fn V
14		funfvex 5712	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑀 ∈ dom Base) → (Base‘𝑀) ∈ V)
15	14	funfni 5483	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑀 ∈ V) → (Base‘𝑀) ∈ V)
16	13, 12, 15	sylancr 418	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → (Base‘𝑀) ∈ V)
17	16	pwexd 4318	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → 𝒫 (Base‘𝑀) ∈ V)
18	17	mptexd 5944	. . . 4 ⊢ (𝑀 ∈ 𝑉 → (𝑠 ∈ 𝒫 (Base‘𝑀) ↦ {𝑥 ∈ (Base‘𝑀) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}) ∈ V)
19	2, 11, 12, 18	fvmptd3 5799	. . 3 ⊢ (𝑀 ∈ 𝑉 → (Cntz‘𝑀) = (𝑠 ∈ 𝒫 (Base‘𝑀) ↦ {𝑥 ∈ (Base‘𝑀) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}))
20	19, 18	eqeltrd 2315	. 2 ⊢ (𝑀 ∈ 𝑉 → (Cntz‘𝑀) ∈ V)
21	1, 20	eqeltrid 2325	1 ⊢ (𝑀 ∈ 𝑉 → 𝑍 ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821  𝒫 cpw 3688   ↦ cmpt 4192   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085  Basecbs 13404  +_gcplusg 13484  Cntzccntz 14140

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8271  ax-resscn 8272  ax-1re 8274  ax-addrcl 8277

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-inn 9308  df-ndx 13407  df-slot 13408  df-base 13410  df-cntz 14142

This theorem is used by:  cntrval  14145