Theorem elkgen 20606

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 20605	. . 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 2525	. 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 3639	. . . . . . 7 |- ( x = A -> ( x i^i k ) = ( A i^i k ) )
4	3	eleq1d 2524	. . . . . 6 |- ( x = A -> ( ( x i^i k ) e. ( J ↾t k ) <-> ( A i^i k ) e. ( J ↾t k ) ) )
5	4	imbi2d 322	. . . . 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 2839	. . . 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 3208	. . 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 19998	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
9		elpw2g 4583	. . . . 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 716	. . 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 265	. 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 261	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 189   /\ wa 375   = wceq 1455   e. wcel 1898   A.wral 2749   {crab 2753   i^i cin 3415   C_ wss 3416   ~Pcpw 3963   ` cfv 5605  (class class class)co 6320   ↾_t crest 15374  TopOnctopon 19973   Compccmp 20456  𝑘Genckgen 20603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-iota 5569  df-fun 5607  df-fv 5613  df-ov 6323  df-top 19976  df-topon 19978  df-kgen 20604

This theorem is referenced by:  kgeni  20607  kgentopon  20608  kgenss  20613  kgenidm  20617  iskgen3  20619  kgen2ss  20625  kgencn  20626  kgencn3  20628  txkgen  20722