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Theorem cntzsnval 17697
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 + = (+g𝑀)
cntzfval.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntzsnval (𝑌𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ (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.
StepHypRef Expression
1 snssi 4315 . . 3 (𝑌𝐵 → {𝑌} ⊆ 𝐵)
2 cntzfval.b . . . 4 𝐵 = (Base‘𝑀)
3 cntzfval.p . . . 4 + = (+g𝑀)
4 cntzfval.z . . . 4 𝑍 = (Cntz‘𝑀)
52, 3, 4cntzval 17694 . . 3 ({𝑌} ⊆ 𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)})
61, 5syl 17 . 2 (𝑌𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)})
7 oveq2 6623 . . . . 5 (𝑦 = 𝑌 → (𝑥 + 𝑦) = (𝑥 + 𝑌))
8 oveq1 6622 . . . . 5 (𝑦 = 𝑌 → (𝑦 + 𝑥) = (𝑌 + 𝑥))
97, 8eqeq12d 2636 . . . 4 (𝑦 = 𝑌 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
109ralsng 4196 . . 3 (𝑌𝐵 → (∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
1110rabbidv 3181 . 2 (𝑌𝐵 → {𝑥𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
126, 11eqtrd 2655 1 (𝑌𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2908  {crab 2912  wss 3560  {csn 4155  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  Cntzccntz 17688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-cntz 17690
This theorem is referenced by:  elcntzsn  17698  cntziinsn  17707
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