Theorem tskmcl 10755

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 10753	. . 3 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2		ssrab2 4011	. . . 4 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski
3		id 22	. . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
4		grothtsk 10749	. . . . . . 7 ⊢ ∪ Tarski = V
5	3, 4	eleqtrrdi 2850	. . . . . 6 ⊢ (𝐴 ∈ V → 𝐴 ∈ ∪ Tarski)
6		eluni2 4842	. . . . . 6 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
7	5, 6	sylib 219	. . . . 5 ⊢ (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
8		rabn0 4317	. . . . 5 ⊢ ({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
9	7, 8	sylibr 235	. . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅)
10		inttsk 10688	. . . 4 ⊢ (({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski ∧ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅) → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski)
11	2, 9, 10	sylancr 593	. . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski)
12	1, 11	eqeltrd 2839	. 2 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
13		fvprc 6819	. . 3 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅)
14		0tsk 10669	. . 3 ⊢ ∅ ∈ Tarski
15	13, 14	eqeltrdi 2847	. 2 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
16	12, 15	pm2.61i 183	1 ⊢ (tarskiMap‘𝐴) ∈ Tarski

Syntax hints:  ¬ wn 3   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063  {crab 3391  Vcvv 3431   ⊆ wss 3883  ∅c0 4261  ∪ cuni 4838  ∩ cint 4877  ‘cfv 6485  Tarskictsk 10662  tarskiMapctskm 10751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-groth 10737

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-er 8633  df-en 8884  df-dom 8885  df-tsk 10663  df-tskm 10752

This theorem is referenced by:  eltskm  10757