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Theorem tskmcl 10881
Description: A Tarski class that contains 𝐴 is a Tarski class. (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)
Assertion
Ref Expression
tskmcl (tarskiMap‘𝐴) ∈ Tarski

Proof of Theorem tskmcl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tskmval 10879 . . 3 (𝐴 ∈ V → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
2 ssrab2 4080 . . . 4 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ⊆ Tarski
3 id 22 . . . . . . 7 (𝐴 ∈ V → 𝐴 ∈ V)
4 grothtsk 10875 . . . . . . 7 Tarski = V
53, 4eleqtrrdi 2852 . . . . . 6 (𝐴 ∈ V → 𝐴 Tarski)
6 eluni2 4911 . . . . . 6 (𝐴 Tarski ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
75, 6sylib 218 . . . . 5 (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴𝑥)
8 rabn0 4389 . . . . 5 ({𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
97, 8sylibr 234 . . . 4 (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅)
10 inttsk 10814 . . . 4 (({𝑥 ∈ Tarski ∣ 𝐴𝑥} ⊆ Tarski ∧ {𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅) → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ Tarski)
112, 9, 10sylancr 587 . . 3 (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ Tarski)
121, 11eqeltrd 2841 . 2 (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
13 fvprc 6898 . . 3 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅)
14 0tsk 10795 . . 3 ∅ ∈ Tarski
1513, 14eqeltrdi 2849 . 2 𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
1612, 15pm2.61i 182 1 (tarskiMap‘𝐴) ∈ Tarski
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2108  wne 2940  wrex 3070  {crab 3436  Vcvv 3480  wss 3951  c0 4333   cuni 4907   cint 4946  cfv 6561  Tarskictsk 10788  tarskiMapctskm 10877
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-groth 10863
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-er 8745  df-en 8986  df-dom 8987  df-tsk 10789  df-tskm 10878
This theorem is referenced by:  eltskm  10883
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