Theorem cntzsnval 19355

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 4813	. . 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 19352	. . 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 2751	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
10	9	ralsng 4680	. . 3 ⊢ (𝑌 ∈ 𝐵 → (∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
11	10	rabbidv 3441	. 2 ⊢ (𝑌 ∈ 𝐵 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
12	6, 11	eqtrd 2775	1 ⊢ (𝑌 ∈ 𝐵 → (𝑍‘{𝑌}) = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433   ⊆ wss 3963  {csn 4631  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  Cntzccntz 19346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-cntz 19348

This theorem is referenced by:  elcntzsn  19356  cntziinsn  19368