Theorem kgenval 21338

Description: Value of the compact generator. (The "k" in 𝑘Gen comes from the name "k-space" for these spaces, after the German word kompakt.) (Contributed by Mario Carneiro, 20-Mar-2015.)

Assertion

Ref	Expression
kgenval	|- ( 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 ) ) } )

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

Proof of Theorem kgenval

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-kgen 21337	. . 3 |- 𝑘Gen = ( j e. Top |-> { x e. ~P U. j | A. k e. ~P U. j ( ( j ↾t k ) e. Comp -> ( x i^i k ) e. ( j ↾t k ) ) } )
2	1	a1i 11	. 2 |- ( J e. (TopOn ` X ) -> 𝑘Gen = ( j e. Top |-> { x e. ~P U. j | A. k e. ~P U. j ( ( j ↾t k ) e. Comp -> ( x i^i k ) e. ( j ↾t k ) ) } ) )
3		unieq 4444	. . . . 5 |- ( j = J -> U. j = U. J )
4		toponuni 20719	. . . . . 6 |- ( J e. (TopOn ` X ) -> X = U. J )
5	4	eqcomd 2628	. . . . 5 |- ( J e. (TopOn ` X ) -> U. J = X )
6	3, 5	sylan9eqr 2678	. . . 4 |- ( ( J e. (TopOn ` X ) /\ j = J ) -> U. j = X )
7	6	pweqd 4163	. . 3 |- ( ( J e. (TopOn ` X ) /\ j = J ) -> ~P U. j = ~P X )
8		oveq1 6657	. . . . . . 7 |- ( j = J -> ( j ↾t k ) = ( J ↾t k ) )
9	8	eleq1d 2686	. . . . . 6 |- ( j = J -> ( ( j ↾t k ) e. Comp <-> ( J ↾t k ) e. Comp ) )
10	8	eleq2d 2687	. . . . . 6 |- ( j = J -> ( ( x i^i k ) e. ( j ↾t k ) <-> ( x i^i k ) e. ( J ↾t k ) ) )
11	9, 10	imbi12d 334	. . . . 5 |- ( j = J -> ( ( ( j ↾t k ) e. Comp -> ( x i^i k ) e. ( j ↾t k ) ) <-> ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) )
12	11	adantl 482	. . . 4 |- ( ( J e. (TopOn ` X ) /\ j = J ) -> ( ( ( j ↾t k ) e. Comp -> ( x i^i k ) e. ( j ↾t k ) ) <-> ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) )
13	7, 12	raleqbidv 3152	. . 3 |- ( ( J e. (TopOn ` X ) /\ j = J ) -> ( A. k e. ~P U. j ( ( j ↾t k ) e. Comp -> ( x i^i k ) e. ( j ↾t k ) ) <-> A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) )
14	7, 13	rabeqbidv 3195	. 2 |- ( ( J e. (TopOn ` X ) /\ j = J ) -> { x e. ~P U. j | A. k e. ~P U. j ( ( j ↾t k ) e. Comp -> ( x i^i k ) e. ( j ↾t k ) ) } = { x e. ~P X | A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) } )
15		topontop 20718	. 2 |- ( J e. (TopOn ` X ) -> J e. Top )
16		toponmax 20730	. . 3 |- ( J e. (TopOn ` X ) -> X e. J )
17		pwexg 4850	. . 3 |- ( X e. J -> ~P X e. _V )
18		rabexg 4812	. . 3 |- ( ~P X e. _V -> { x e. ~P X | A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) } e. _V )
19	16, 17, 18	3syl 18	. 2 |- ( J e. (TopOn ` X ) -> { x e. ~P X | A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) } e. _V )
20	2, 14, 15, 19	fvmptd 6288	1 |- ( 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 ) ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   i^i cin 3573   ~Pcpw 4158   U.cuni 4436   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  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:  elkgen  21339  kgentopon  21341