Theorem cntzsnval 19279

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 4807	. . 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 19276	. . 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 7424	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑥 + 𝑦) = (𝑥 + 𝑌))
8		oveq1 7423	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 + 𝑥) = (𝑌 + 𝑥))
9	7, 8	eqeq12d 2741	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
10	9	ralsng 4673	. . 3 ⊢ (𝑌 ∈ 𝐵 → (∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
11	10	rabbidv 3427	. 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 1533   ∈ wcel 2098  ∀wral 3051  {crab 3419   ⊆ wss 3939  {csn 4624  ‘cfv 6543  (class class class)co 7416  Basecbs 17179  +_gcplusg 17232  Cntzccntz 19270

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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-cntz 19272

This theorem is referenced by:  elcntzsn  19280  cntziinsn  19292