Theorem sstskm 9664

Description: Being a part of ( tarskiMap ` A ). (Contributed by FL, 17-Apr-2011.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)

Assertion

Ref	Expression
sstskm	|- ( A e. V -> ( B C_ ( tarskiMap ` A ) <-> A. x e. Tarski ( A e. x -> B C_ x ) ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem sstskm

Step	Hyp	Ref	Expression
1		tskmval 9661	. . . 4 |- ( A e. V -> ( tarskiMap ` A ) = |^| { x e. Tarski | A e. x } )
2		df-rab 2921	. . . . 5 |- { x e. Tarski | A e. x } = { x | ( x e. Tarski /\ A e. x ) }
3	2	inteqi 4479	. . . 4 |- |^| { x e. Tarski | A e. x } = |^| { x | ( x e. Tarski /\ A e. x ) }
4	1, 3	syl6eq 2672	. . 3 |- ( A e. V -> ( tarskiMap ` A ) = |^| { x | ( x e. Tarski /\ A e. x ) } )
5	4	sseq2d 3633	. 2 |- ( A e. V -> ( B C_ ( tarskiMap ` A ) <-> B C_ |^| { x | ( x e. Tarski /\ A e. x ) } ) )
6		impexp 462	. . . 4 |- ( ( ( x e. Tarski /\ A e. x ) -> B C_ x ) <-> ( x e. Tarski -> ( A e. x -> B C_ x ) ) )
7	6	albii 1747	. . 3 |- ( A. x ( ( x e. Tarski /\ A e. x ) -> B C_ x ) <-> A. x ( x e. Tarski -> ( A e. x -> B C_ x ) ) )
8		ssintab 4494	. . 3 |- ( B C_ |^| { x | ( x e. Tarski /\ A e. x ) } <-> A. x ( ( x e. Tarski /\ A e. x ) -> B C_ x ) )
9		df-ral 2917	. . 3 |- ( A. x e. Tarski ( A e. x -> B C_ x ) <-> A. x ( x e. Tarski -> ( A e. x -> B C_ x ) ) )
10	7, 8, 9	3bitr4i 292	. 2 |- ( B C_ |^| { x | ( x e. Tarski /\ A e. x ) } <-> A. x e. Tarski ( A e. x -> B C_ x ) )
11	5, 10	syl6bb 276	1 |- ( A e. V -> ( B C_ ( tarskiMap ` A ) <-> A. x e. Tarski ( A e. x -> B C_ x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   C_ wss 3574   |^|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: (None)