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Theorem tskmcl 9847
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 9845 . . 3 (𝐴 ∈ V → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
2 ssrab2 3820 . . . 4 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ⊆ Tarski
3 id 22 . . . . . . 7 (𝐴 ∈ V → 𝐴 ∈ V)
4 grothtsk 9841 . . . . . . 7 Tarski = V
53, 4syl6eleqr 2842 . . . . . 6 (𝐴 ∈ V → 𝐴 Tarski)
6 eluni2 4584 . . . . . 6 (𝐴 Tarski ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
75, 6sylib 208 . . . . 5 (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴𝑥)
8 rabn0 4093 . . . . 5 ({𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
97, 8sylibr 224 . . . 4 (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅)
10 inttsk 9780 . . . 4 (({𝑥 ∈ Tarski ∣ 𝐴𝑥} ⊆ Tarski ∧ {𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅) → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ Tarski)
112, 9, 10sylancr 698 . . 3 (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ∈ Tarski)
121, 11eqeltrd 2831 . 2 (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
13 fvprc 6338 . . 3 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅)
14 0tsk 9761 . . 3 ∅ ∈ Tarski
1513, 14syl6eqel 2839 . 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 2131  wne 2924  wrex 3043  {crab 3046  Vcvv 3332  wss 3707  c0 4050   cuni 4580   cint 4619  cfv 6041  Tarskictsk 9754  tarskiMapctskm 9843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-groth 9829
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-int 4620  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-er 7903  df-en 8114  df-dom 8115  df-tsk 9755  df-tskm 9844
This theorem is referenced by:  eltskm  9849
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