Theorem elkgen 19236

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 19235	. . 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 2522	. 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 3648	. . . . . . 7 |- ( x = A -> ( x i^i k ) = ( A i^i k ) )
4	3	eleq1d 2521	. . . . . 6 |- ( x = A -> ( ( x i^i k ) e. ( J ↾t k ) <-> ( A i^i k ) e. ( J ↾t k ) ) )
5	4	imbi2d 316	. . . . 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 2843	. . . 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 3218	. . 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 18660	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
9		elpw2g 4558	. . . . 5 |- ( X e. J -> ( A e. ~P X <-> A C_ X ) )
10	8, 9	syl 16	. . . 4 |- ( J e. (TopOn ` X ) -> ( A e. ~P X <-> A C_ X ) )
11	10	anbi1d 704	. . 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 257	. 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 253	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 184   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2796   {crab 2800   i^i cin 3430   C_ wss 3431   ~Pcpw 3963   ` cfv 5521  (class class class)co 6195   ↾_t crest 14473  TopOnctopon 18626   Compccmp 19116  𝑘Genckgen 19233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-ov 6198  df-top 18630  df-topon 18633  df-kgen 19234

This theorem is referenced by:  kgeni  19237  kgentopon  19238  kgenss  19243  kgenidm  19247  iskgen3  19249  kgen2ss  19255  kgencn  19256  kgencn3  19258  txkgen  19352