Theorem cntzrcl 19240

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 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 4325	. . . 4 ⊢ ¬ 𝑋 ∈ ∅
2		cntzrcl.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝑀)
3		fvprc 6876	. . . . . . . 8 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
4	2, 3	eqtrid 2778	. . . . . . 7 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅)
5	4	fveq1d 6886	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = (∅‘𝑆))
6		0fv 6928	. . . . . 6 ⊢ (∅‘𝑆) = ∅
7	5, 6	eqtrdi 2782	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = ∅)
8	7	eleq2d 2813	. . . 4 ⊢ (¬ 𝑀 ∈ V → (𝑋 ∈ (𝑍‘𝑆) ↔ 𝑋 ∈ ∅))
9	1, 8	mtbiri 327	. . 3 ⊢ (¬ 𝑀 ∈ V → ¬ 𝑋 ∈ (𝑍‘𝑆))
10	9	con4i 114	. 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑀 ∈ V)
11		cntzrcl.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑀)
12		eqid 2726	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
13	11, 12, 2	cntzfval 19233	. . . . . . 7 ⊢ (𝑀 ∈ V → 𝑍 = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}))
14	10, 13	syl 17	. . . . . 6 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑍 = (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}))
15	14	dmeqd 5898	. . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 = dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}))
16		eqid 2726	. . . . . 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)})
17	16	dmmptss 6233	. . . . 5 ⊢ dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}) ⊆ 𝒫 𝐵
18	15, 17	eqsstrdi 4031	. . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 ⊆ 𝒫 𝐵)
19		elfvdm 6921	. . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ dom 𝑍)
20	18, 19	sseldd 3978	. . 3 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ 𝒫 𝐵)
21	20	elpwid 4606	. 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ⊆ 𝐵)
22	10, 21	jca 511	1 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055  {crab 3426  Vcvv 3468   ⊆ wss 3943  ∅c0 4317  𝒫 cpw 4597   ↦ cmpt 5224  dom cdm 5669  ‘cfv 6536  (class class class)co 7404  Basecbs 17150  +_gcplusg 17203  Cntzccntz 19228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-cntz 19230

This theorem is referenced by:  cntzssv  19241  cntzi  19242  resscntz  19246  cntzmhm  19254  oppgcntz  19280