Theorem tskmval 9661

Description: Value of our tarski map. (Contributed by FL, 30-Dec-2010.) (Revised by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
tskmval	|- ( A e. V -> ( tarskiMap ` A ) = |^| { x e. Tarski | A e. x } )

Distinct variable group:   x, A

Allowed substitution hint:   V( x)

Proof of Theorem tskmval

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. V -> A e. _V )
2		grothtsk 9657	. . . . 5 |- U. Tarski = _V
3	1, 2	syl6eleqr 2712	. . . 4 |- ( A e. V -> A e. U. Tarski )
4		eluni2 4440	. . . 4 |- ( A e. U. Tarski <-> E. x e. Tarski A e. x )
5	3, 4	sylib 208	. . 3 |- ( A e. V -> E. x e. Tarski A e. x )
6		intexrab 4823	. . 3 |- ( E. x e. Tarski A e. x <-> |^| { x e. Tarski | A e. x } e. _V )
7	5, 6	sylib 208	. 2 |- ( A e. V -> |^| { x e. Tarski | A e. x } e. _V )
8		eleq1 2689	. . . . 5 |- ( y = A -> ( y e. x <-> A e. x ) )
9	8	rabbidv 3189	. . . 4 |- ( y = A -> { x e. Tarski | y e. x } = { x e. Tarski | A e. x } )
10	9	inteqd 4480	. . 3 |- ( y = A -> |^| { x e. Tarski | y e. x } = |^| { x e. Tarski | A e. x } )
11		df-tskm 9660	. . 3 |- tarskiMap = ( y e. _V |-> |^| { x e. Tarski | y e. x } )
12	10, 11	fvmptg 6280	. 2 |- ( ( A e. _V /\ |^| { x e. Tarski | A e. x } e. _V ) -> ( tarskiMap ` A ) = |^| { x e. Tarski | A e. x } )
13	1, 7, 12	syl2anc 693	1 |- ( A e. V -> ( tarskiMap ` A ) = |^| { x e. Tarski | A e. x } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   U.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-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:  tskmid  9662  tskmcl  9663  sstskm  9664  eltskm  9665