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Theorem tskmcl 10301
 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 10299 . . 3 (𝐴 ∈ V → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
2 ssrab2 3984 . . . 4 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ⊆ Tarski
3 id 22 . . . . . . 7 (𝐴 ∈ V → 𝐴 ∈ V)
4 grothtsk 10295 . . . . . . 7 Tarski = V
53, 4eleqtrrdi 2863 . . . . . 6 (𝐴 ∈ V → 𝐴 Tarski)
6 eluni2 4802 . . . . . 6 (𝐴 Tarski ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
75, 6sylib 221 . . . . 5 (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴𝑥)
8 rabn0 4281 . . . . 5 ({𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
97, 8sylibr 237 . . . 4 (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅)
10 inttsk 10234 . . . 4 (({𝑥 ∈ Tarski ∣ 𝐴𝑥} ⊆ Tarski ∧ {𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅) → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ Tarski)
112, 9, 10sylancr 590 . . 3 (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ Tarski)
121, 11eqeltrd 2852 . 2 (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
13 fvprc 6650 . . 3 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅)
14 0tsk 10215 . . 3 ∅ ∈ Tarski
1513, 14eqeltrdi 2860 . 2 𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
1612, 15pm2.61i 185 1 (tarskiMap‘𝐴) ∈ Tarski
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3858  ∅c0 4225  ∪ cuni 4798  ∩ cint 4838  ‘cfv 6335  Tarskictsk 10208  tarskiMapctskm 10297 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-groth 10283 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-int 4839  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-er 8299  df-en 8528  df-dom 8529  df-tsk 10209  df-tskm 10298 This theorem is referenced by:  eltskm  10303
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