Theorem elkgen 21339

Description: Value of the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)

Assertion

Ref	Expression
elkgen	|- ( J e. (TopOn ` X ) -> ( A e. (𝑘Gen ` J ) <-> ( A C_ X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( A i^i k ) e. ( J ↾t k ) ) ) ) )

Distinct variable groups:   A, k   k, J   k, X

Proof of Theorem elkgen

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		kgenval 21338	. . 3 |- ( J e. (TopOn ` X ) -> (𝑘Gen ` J ) = { x e. ~P X | A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) } )
2	1	eleq2d 2687	. 2 |- ( J e. (TopOn ` X ) -> ( A e. (𝑘Gen ` J ) <-> A e. { x e. ~P X | A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) } ) )
3		ineq1 3807	. . . . . . 7 |- ( x = A -> ( x i^i k ) = ( A i^i k ) )
4	3	eleq1d 2686	. . . . . 6 |- ( x = A -> ( ( x i^i k ) e. ( J ↾t k ) <-> ( A i^i k ) e. ( J ↾t k ) ) )
5	4	imbi2d 330	. . . . 5 |- ( x = A -> ( ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) <-> ( ( J ↾t k ) e. Comp -> ( A i^i k ) e. ( J ↾t k ) ) ) )
6	5	ralbidv 2986	. . . 4 |- ( x = A -> ( A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) <-> A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( A i^i k ) e. ( J ↾t k ) ) ) )
7	6	elrab 3363	. . 3 |- ( A e. { x e. ~P X | A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) } <-> ( A e. ~P X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( A i^i k ) e. ( J ↾t k ) ) ) )
8		toponmax 20730	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
9		elpw2g 4827	. . . . 5 |- ( X e. J -> ( A e. ~P X <-> A C_ X ) )
10	8, 9	syl 17	. . . 4 |- ( J e. (TopOn ` X ) -> ( A e. ~P X <-> A C_ X ) )
11	10	anbi1d 741	. . 3 |- ( J e. (TopOn ` X ) -> ( ( A e. ~P X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( A i^i k ) e. ( J ↾t k ) ) ) <-> ( A C_ X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( A i^i k ) e. ( J ↾t k ) ) ) ) )
12	7, 11	syl5bb 272	. 2 |- ( J e. (TopOn ` X ) -> ( A e. { x e. ~P X | A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) } <-> ( A C_ X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( A i^i k ) e. ( J ↾t k ) ) ) ) )
13	2, 12	bitrd 268	1 |- ( J e. (TopOn ` X ) -> ( A e. (𝑘Gen ` J ) <-> ( A C_ X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( A i^i k ) e. ( J ↾t k ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081  TopOnctopon 20715   Compccmp 21189  𝑘Genckgen 21336

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-pow 4843  ax-pr 4906  ax-un 6949

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-csb 3534  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-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-ov 6653  df-top 20699  df-topon 20716  df-kgen 21337

This theorem is referenced by:  kgeni  21340  kgentopon  21341  kgenss  21346  kgenidm  21350  iskgen3  21352  kgen2ss  21358  kgencn  21359  kgencn3  21361  txkgen  21455