Theorem tskmcl 10910

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.

Step	Hyp	Ref	Expression
1		tskmval 10908	. . 3 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2		ssrab2 4103	. . . 4 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski
3		id 22	. . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
4		grothtsk 10904	. . . . . . 7 ⊢ ∪ Tarski = V
5	3, 4	eleqtrrdi 2855	. . . . . 6 ⊢ (𝐴 ∈ V → 𝐴 ∈ ∪ Tarski)
6		eluni2 4935	. . . . . 6 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
7	5, 6	sylib 218	. . . . 5 ⊢ (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
8		rabn0 4412	. . . . 5 ⊢ ({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
9	7, 8	sylibr 234	. . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅)
10		inttsk 10843	. . . 4 ⊢ (({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski ∧ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅) → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski)
11	2, 9, 10	sylancr 586	. . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski)
12	1, 11	eqeltrd 2844	. 2 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
13		fvprc 6912	. . 3 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅)
14		0tsk 10824	. . 3 ⊢ ∅ ∈ Tarski
15	13, 14	eqeltrdi 2852	. 2 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
16	12, 15	pm2.61i 182	1 ⊢ (tarskiMap‘𝐴) ∈ Tarski

Syntax hints:  ¬ wn 3   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  {crab 3443  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  ∩ cint 4970  ‘cfv 6573  Tarskictsk 10817  tarskiMapctskm 10906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-groth 10892

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-er 8763  df-en 9004  df-dom 9005  df-tsk 10818  df-tskm 10907

This theorem is referenced by:  eltskm  10912