Theorem tskmcl 9663

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 9661	. . 3 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
2		ssrab2 3687	. . . 4 ⊢ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski
3		id 22	. . . . . . 7 ⊢ (𝐴 ∈ V → 𝐴 ∈ V)
4		grothtsk 9657	. . . . . . 7 ⊢ ∪ Tarski = V
5	3, 4	syl6eleqr 2712	. . . . . 6 ⊢ (𝐴 ∈ V → 𝐴 ∈ ∪ Tarski)
6		eluni2 4440	. . . . . 6 ⊢ (𝐴 ∈ ∪ Tarski ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
7	5, 6	sylib 208	. . . . 5 ⊢ (𝐴 ∈ V → ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
8		rabn0 3958	. . . . 5 ⊢ ({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Tarski 𝐴 ∈ 𝑥)
9	7, 8	sylibr 224	. . . 4 ⊢ (𝐴 ∈ V → {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅)
10		inttsk 9596	. . . 4 ⊢ (({𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ⊆ Tarski ∧ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ≠ ∅) → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski)
11	2, 9, 10	sylancr 695	. . 3 ⊢ (𝐴 ∈ V → ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ∈ Tarski)
12	1, 11	eqeltrd 2701	. 2 ⊢ (𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
13		fvprc 6185	. . 3 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) = ∅)
14		0tsk 9577	. . 3 ⊢ ∅ ∈ Tarski
15	13, 14	syl6eqel 2709	. 2 ⊢ (¬ 𝐴 ∈ V → (tarskiMap‘𝐴) ∈ Tarski)
16	12, 15	pm2.61i 176	1 ⊢ (tarskiMap‘𝐴) ∈ Tarski

Syntax hints:  ¬ wn 3   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∪ cuni 4436  ∩ cint 4475  ‘cfv 5888  Tarskictsk 9570  tarskiMapctskm 9659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-groth 9645

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-tsk 9571  df-tskm 9660

This theorem is referenced by:  eltskm  9665