Theorem cntzsnval 19263

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 4775	. . 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 19260	. . 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 7398	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑥 + 𝑦) = (𝑥 + 𝑌))
8		oveq1 7397	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 + 𝑥) = (𝑌 + 𝑥))
9	7, 8	eqeq12d 2746	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
10	9	ralsng 4642	. . 3 ⊢ (𝑌 ∈ 𝐵 → (∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
11	10	rabbidv 3416	. 2 ⊢ (𝑌 ∈ 𝐵 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
12	6, 11	eqtrd 2765	1 ⊢ (𝑌 ∈ 𝐵 → (𝑍‘{𝑌}) = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408   ⊆ wss 3917  {csn 4592  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  Cntzccntz 19254

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-cntz 19256

This theorem is referenced by:  elcntzsn  19264  cntziinsn  19276