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Theorem cntzsnval 18107
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 4557 . . 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 18104 . . 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 6913 . . . . 5 (𝑦 = 𝑌 → (𝑥 + 𝑦) = (𝑥 + 𝑌))
8 oveq1 6912 . . . . 5 (𝑦 = 𝑌 → (𝑦 + 𝑥) = (𝑌 + 𝑥))
97, 8eqeq12d 2840 . . . 4 (𝑦 = 𝑌 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
109ralsng 4438 . . 3 (𝑌𝐵 → (∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
1110rabbidv 3402 . 2 (𝑌𝐵 → {𝑥𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
126, 11eqtrd 2861 1 (𝑌𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166  wral 3117  {crab 3121  wss 3798  {csn 4397  cfv 6123  (class class class)co 6905  Basecbs 16222  +gcplusg 16305  Cntzccntz 18098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-cntz 18100
This theorem is referenced by:  elcntzsn  18108  cntziinsn  18117
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