Theorem tskmcl 9542

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 9540	. . 3 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2		ssrab2 3650	. . . 4 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski
3		id 22	. . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
4		grothtsk 9536	. . . . . . 7 ⊢ ∪ Tarski = V
5	3, 4	syl6eleqr 2699	. . . . . 6 ⊢ (𝐴 ∈ V → 𝐴 ∈ ∪ Tarski)
6		eluni2 4376	. . . . . 6 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
7	5, 6	sylib 207	. . . . 5 ⊢ (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
8		rabn0 3912	. . . . 5 ⊢ ({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
9	7, 8	sylibr 223	. . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅)
10		inttsk 9475	. . . 4 ⊢ (({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski ∧ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅) → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski)
11	2, 9, 10	sylancr 694	. . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski)
12	1, 11	eqeltrd 2688	. 2 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
13		fvprc 6097	. . 3 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅)
14		0tsk 9456	. . 3 ⊢ ∅ ∈ Tarski
15	13, 14	syl6eqel 2696	. 2 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
16	12, 15	pm2.61i 175	1 ⊢ (tarskiMap‘𝐴) ∈ Tarski

Syntax hints:  ¬ wn 3   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ∩ cint 4410  ‘cfv 5804  Tarskictsk 9449  tarskiMapctskm 9538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-groth 9524

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-tsk 9450  df-tskm 9539

This theorem is referenced by:  eltskm  9544