Theorem tskmid 9662

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 e. V -> A e. ( tarskiMap ` A ) )

Proof of Theorem tskmid

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( A e. x -> A e. x )
2	1	rgenw 2924	. . 3 |- A. x e. Tarski ( A e. x -> A e. x )
3		elintrabg 4489	. . 3 |- ( A e. V -> ( A e. |^| { x e. Tarski | A e. x } <-> A. x e. Tarski ( A e. x -> A e. x ) ) )
4	2, 3	mpbiri 248	. 2 |- ( A e. V -> A e. |^| { x e. Tarski | A e. x } )
5		tskmval 9661	. 2 |- ( A e. V -> ( tarskiMap ` A ) = |^| { x e. Tarski | A e. x } )
6	4, 5	eleqtrrd 2704	1 |- ( A e. V -> A e. ( tarskiMap ` A ) )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   {crab 2916   |^|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-pr 4906  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-iota 5851  df-fun 5890  df-fv 5896  df-tsk 9571  df-tskm 9660

This theorem is referenced by:  eltskm  9665