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Theorem tskmcl 10776
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 10774 . . 3 (𝐴 ∈ V → (tarskiMap‘𝐴) = {𝑥 ∈ Tarski ∣ 𝐴𝑥})
2 ssrab2 4037 . . . 4 {𝑥 ∈ Tarski ∣ 𝐴𝑥} ⊆ Tarski
3 id 22 . . . . . . 7 (𝐴 ∈ V → 𝐴 ∈ V)
4 grothtsk 10770 . . . . . . 7 Tarski = V
53, 4eleqtrrdi 2849 . . . . . 6 (𝐴 ∈ V → 𝐴 Tarski)
6 eluni2 4869 . . . . . 6 (𝐴 Tarski ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
75, 6sylib 217 . . . . 5 (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴𝑥)
8 rabn0 4345 . . . . 5 ({𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴𝑥)
97, 8sylibr 233 . . . 4 (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴𝑥} ≠ ∅)
10 inttsk 10709 . . . 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 2838 . 2 (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
13 fvprc 6834 . . 3 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅)
14 0tsk 10690 . . 3 ∅ ∈ Tarski
1513, 14eqeltrdi 2846 . 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 2106  wne 2943  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  c0 4282   cuni 4865   cint 4907  cfv 6496  Tarskictsk 10683  tarskiMapctskm 10772
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671  ax-groth 10758
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-er 8647  df-en 8883  df-dom 8884  df-tsk 10684  df-tskm 10773
This theorem is referenced by:  eltskm  10778
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