Theorem tskmid 10817

Description: The set 𝐴 is an element of the smallest Tarski class that contains 𝐴. CLASSES1 th. 5. (Contributed by FL, 30-Dec-2010.) (Proof shortened by Mario Carneiro, 21-Sep-2014.)

Assertion

Ref	Expression
tskmid	⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))

Proof of Theorem tskmid

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
2	1	rgenw 3064	. . 3 ⊢ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
3		elintrabg 4958	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)))
4	2, 3	mpbiri 257	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
5		tskmval 10816	. 2 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
6	4, 5	eleqtrrd 2835	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))

Syntax hints:   → wi 4   ∈ wcel 2106  ∀wral 3060  {crab 3431  ∩ cint 4943  ‘cfv 6532  Tarskictsk 10725  tarskiMapctskm 10814

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 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-groth 10800

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-tsk 10726  df-tskm 10815

This theorem is referenced by:  eltskm  10820