Theorem cntzrcl 19369

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 4292	. . . 4 ⊢ ¬ 𝑋 ∈ ∅
2		cntzrcl.z	. . . . . . . 8 ⊢ 𝑍 = (Cntz‘𝑀)
3		fvprc 6861	. . . . . . . 8 ⊢ (¬ 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
4	2, 3	eqtrid 2811	. . . . . . 7 ⊢ (¬ 𝑀 ∈ V → 𝑍 = ∅)
5	4	fveq1d 6871	. . . . . 6 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = (∅‘𝑆))
6		0fv 6910	. . . . . 6 ⊢ (∅‘𝑆) = ∅
7	5, 6	eqtrdi 2815	. . . . 5 ⊢ (¬ 𝑀 ∈ V → (𝑍‘𝑆) = ∅)
8	7	eleq2d 2850	. . . 4 ⊢ (¬ 𝑀 ∈ V → (𝑋 ∈ (𝑍‘𝑆) ↔ 𝑋 ∈ ∅))
9	1, 8	mtbiri 329	. . 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 2764	. . . . . . . 8 ⊢ (+g‘𝑀) = (+g‘𝑀)
13	11, 12, 2	cntzfval 19362	. . . . . . 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 5883	. . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 = dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}))
16		eqid 2764	. . . . . 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 6230	. . . . 5 ⊢ dom (𝑥 ∈ 𝒫 𝐵 ↦ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝑥 (𝑦(+g‘𝑀)𝑧) = (𝑧(+g‘𝑀)𝑦)}) ⊆ 𝒫 𝐵
18	15, 17	eqsstrdi 3982	. . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → dom 𝑍 ⊆ 𝒫 𝐵)
19		elfvdm 6903	. . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ dom 𝑍)
20	18, 19	sseldd 3939	. . 3 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ∈ 𝒫 𝐵)
21	20	elpwid 4566	. 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ⊆ 𝐵)
22	10, 21	jca 519	1 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416  Vcvv 3456   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4557   ↦ cmpt 5183  dom cdm 5649  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  +_gcplusg 17288  Cntzccntz 19357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-cntz 19359

This theorem is referenced by:  cntzssv  19370  cntzi  19371  resscntz  19375  cntzmhm  19383  oppgcntz  19406