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Theorem tskmid 10837
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 β†’ 𝐴 ∈ (tarskiMapβ€˜π΄))

Proof of Theorem tskmid
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝐴 ∈ π‘₯ β†’ 𝐴 ∈ π‘₯)
21rgenw 3063 . . 3 βˆ€π‘₯ ∈ Tarski (𝐴 ∈ π‘₯ β†’ 𝐴 ∈ π‘₯)
3 elintrabg 4964 . . 3 (𝐴 ∈ 𝑉 β†’ (𝐴 ∈ ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯} ↔ βˆ€π‘₯ ∈ Tarski (𝐴 ∈ π‘₯ β†’ 𝐴 ∈ π‘₯)))
42, 3mpbiri 257 . 2 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯})
5 tskmval 10836 . 2 (𝐴 ∈ 𝑉 β†’ (tarskiMapβ€˜π΄) = ∩ {π‘₯ ∈ Tarski ∣ 𝐴 ∈ π‘₯})
64, 5eleqtrrd 2834 1 (𝐴 ∈ 𝑉 β†’ 𝐴 ∈ (tarskiMapβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2104  βˆ€wral 3059  {crab 3430  βˆ© cint 4949  β€˜cfv 6542  Tarskictsk 10745  tarskiMapctskm 10834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-groth 10820
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-tsk 10746  df-tskm 10835
This theorem is referenced by:  eltskm  10840
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