Theorem elkgen 20203

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 20202	. . 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 2524	. 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 3679	. . . . . . 7 |- ( x = A -> ( x i^i k ) = ( A i^i k ) )
4	3	eleq1d 2523	. . . . . 6 |- ( x = A -> ( ( x i^i k ) e. ( J ↾t k ) <-> ( A i^i k ) e. ( J ↾t k ) ) )
5	4	imbi2d 314	. . . . 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 2893	. . . 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 3254	. . 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 19596	. . . . 5 |- ( J e. (TopOn ` X ) -> X e. J )
9		elpw2g 4600	. . . . 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 702	. . 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 367   = wceq 1398   e. wcel 1823   A.wral 2804   {crab 2808   i^i cin 3460   C_ wss 3461   ~Pcpw 3999   ` cfv 5570  (class class class)co 6270   ↾_t crest 14910  TopOnctopon 19562   Compccmp 20053  𝑘Genckgen 20200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-top 19566  df-topon 19569  df-kgen 20201

This theorem is referenced by:  kgeni  20204  kgentopon  20205  kgenss  20210  kgenidm  20214  iskgen3  20216  kgen2ss  20222  kgencn  20223  kgencn3  20225  txkgen  20319