Theorem cntzsnval 19342

Description: Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

Ref	Expression
cntzfval.b	⊢ 𝐵 = (Base‘𝑀)
cntzfval.p	⊢ + = (+g‘𝑀)
cntzfval.z	⊢ 𝑍 = (Cntz‘𝑀)

Assertion

Ref	Expression
cntzsnval	⊢ (𝑌 ∈ 𝐵 → (𝑍‘{𝑌}) = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})

Distinct variable groups:   𝑥, +   𝑥,𝐵   𝑥,𝑀   𝑥,𝑌

Allowed substitution hint:   𝑍(𝑥)

Proof of Theorem cntzsnval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snssi 4808	. . 3 ⊢ (𝑌 ∈ 𝐵 → {𝑌} ⊆ 𝐵)
2		cntzfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
3		cntzfval.p	. . . 4 ⊢ + = (+g‘𝑀)
4		cntzfval.z	. . . 4 ⊢ 𝑍 = (Cntz‘𝑀)
5	2, 3, 4	cntzval 19339	. . 3 ⊢ ({𝑌} ⊆ 𝐵 → (𝑍‘{𝑌}) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)})
6	1, 5	syl 17	. 2 ⊢ (𝑌 ∈ 𝐵 → (𝑍‘{𝑌}) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)})
7		oveq2 7439	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑥 + 𝑦) = (𝑥 + 𝑌))
8		oveq1 7438	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 + 𝑥) = (𝑌 + 𝑥))
9	7, 8	eqeq12d 2753	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
10	9	ralsng 4675	. . 3 ⊢ (𝑌 ∈ 𝐵 → (∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
11	10	rabbidv 3444	. 2 ⊢ (𝑌 ∈ 𝐵 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
12	6, 11	eqtrd 2777	1 ⊢ (𝑌 ∈ 𝐵 → (𝑍‘{𝑌}) = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436   ⊆ wss 3951  {csn 4626  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  Cntzccntz 19333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-cntz 19335

This theorem is referenced by:  elcntzsn  19343  cntziinsn  19355