Theorem tskmid 10964

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 23	. . . 4 ⊢ (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
2	1	rgenw 3084	. . 3 ⊢ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)
3		elintrabg 4926	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥} ↔ ∀𝑥 ∈ Tarski (𝐴 ∈ 𝑥 → 𝐴 ∈ 𝑥)))
4	2, 3	mpbiri 261	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
5		tskmval 10963	. 2 ⊢ (𝐴 ∈ 𝑉 → (tarskiMap‘𝐴) = ∩ {𝑥 ∈ Tarski ∣ 𝐴 ∈ 𝑥})
6	4, 5	eleqtrrd 2867	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (tarskiMap‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ∀wral 3080  {crab 3416  ∩ cint 4912  ‘cfv 6566  Tarskictsk 10872  tarskiMapctskm 10961

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398  ax-groth 10947

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-tsk 10873  df-tskm 10962

This theorem is used by:  eltskm  10967