Theorem cntzrcl 14159

Description: Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Hypotheses

Ref	Expression
cntzrcl.b	⊢ 𝐵 = (Base‘𝑀)
cntzrcl.z	⊢ 𝑍 = (Cntz‘𝑀)

Assertion

Ref	Expression
cntzrcl	⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵))

Proof of Theorem cntzrcl

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

Step	Hyp	Ref	Expression
1		elfvm 5732	. . 3 ⊢ (𝑋 ∈ (𝑍‘𝑆) → ∃𝑗 𝑗 ∈ 𝑍)
2		df-cntz 14148	. . . . . 6 ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
3	2	mptrcl 5791	. . . . 5 ⊢ (𝑗 ∈ (Cntz‘𝑀) → 𝑀 ∈ V)
4		cntzrcl.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝑀)
5	3, 4	eleq2s 2333	. . . 4 ⊢ (𝑗 ∈ 𝑍 → 𝑀 ∈ V)
6	5	exlimiv 1651	. . 3 ⊢ (∃𝑗 𝑗 ∈ 𝑍 → 𝑀 ∈ V)
7	1, 6	syl 14	. 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑀 ∈ V)
8		cntzrcl.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀)
9		eqid 2238	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
10	8, 9, 4	cntzfval 14152	. . . . . . 7 ⊢ (𝑀 ∈ V → 𝑍 = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}))
11	7, 10	syl 14	. . . . . 6 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑍 = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}))
12	11	dmeqd 4985	. . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 = dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}))
13		eqid 2238	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}) = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)})
14	13	dmmptss 5287	. . . . 5 ⊢ dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}) ⊆ 𝒫 𝐵
15	12, 14	eqsstrdi 3300	. . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 ⊆ 𝒫 𝐵)
16		mptrel 4910	. . . . . 6 ⊢ Rel (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)})
17	8, 9, 4	cntzfval 14152	. . . . . . . 8 ⊢ (𝑀 ∈ V → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}))
18	7, 17	syl 14	. . . . . . 7 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)}))
19	18	releqd 4861	. . . . . 6 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (Rel 𝑍 ↔ Rel (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑀)𝑦) = (𝑦(+g‘𝑀)𝑥)})))
20	16, 19	mpbiri 168	. . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → Rel 𝑍)
21		relelfvdm 5730	. . . . 5 ⊢ ((Rel 𝑍 ∧ 𝑋 ∈ (𝑍‘𝑆)) → 𝑆 ∈ dom 𝑍)
22	20, 21	mpancom 426	. . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ dom 𝑍)
23	15, 22	sseldd 3249	. . 3 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ 𝒫 𝐵)
24	23	elpwid 3700	. 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ⊆ 𝐵)
25	7, 24	jca 306	1 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688   ↦ cmpt 4193  dom cdm 4776  Rel wrel 4781  ‘cfv 5380  (class class class)co 6088  Basecbs 13410  +_gcplusg 13490  Cntzccntz 14146

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 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-inn 9314  df-ndx 13413  df-slot 13414  df-base 13416  df-cntz 14148

This theorem is used by:  cntzssv  14160  cntzi  14162  resscntz  14166  cntzmhm  14173