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Theorem tskmcl 9701
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 9699 . . 3 (𝐴 ∈ V → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
2 ssrab2 3720 . . . 4 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ⊆ Tarski
3 id 22 . . . . . . 7 (𝐴 ∈ V → 𝐴 ∈ V)
4 grothtsk 9695 . . . . . . 7 Tarski = V
53, 4syl6eleqr 2741 . . . . . 6 (𝐴 ∈ V → 𝐴 Tarski)
6 eluni2 4472 . . . . . 6 (𝐴 Tarski ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
75, 6sylib 208 . . . . 5 (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴𝑥)
8 rabn0 3991 . . . . 5 ({𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
97, 8sylibr 224 . . . 4 (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅)
10 inttsk 9634 . . . 4 (({𝑥 ∈ Tarski ∣ 𝐴𝑥} ⊆ Tarski ∧ {𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅) → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ Tarski)
112, 9, 10sylancr 696 . . 3 (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ Tarski)
121, 11eqeltrd 2730 . 2 (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
13 fvprc 6223 . . 3 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅)
14 0tsk 9615 . . 3 ∅ ∈ Tarski
1513, 14syl6eqel 2738 . 2 𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
1612, 15pm2.61i 176 1 (tarskiMap‘𝐴) ∈ Tarski
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2030  wne 2823  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  c0 3948   cuni 4468   cint 4507  cfv 5926  Tarskictsk 9608  tarskiMapctskm 9697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-groth 9683
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-er 7787  df-en 7998  df-dom 7999  df-tsk 9609  df-tskm 9698
This theorem is referenced by:  eltskm  9703
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