Theorem iskgen3 21352

Description: Derive the usual definition of "compactly generated". A topology is compactly generated if every subset of X that is open in every compact subset is open. (Contributed by Mario Carneiro, 20-Mar-2015.)

Hypothesis

Ref	Expression
iskgen3.1	|- X = U. J

Assertion

Ref	Expression
iskgen3	|- ( J e. ran 𝑘Gen <-> ( J e. Top /\ A. x e. ~P X ( A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) -> x e. J ) ) )

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

Allowed substitution hint:   X( x)

Proof of Theorem iskgen3

Step	Hyp	Ref	Expression
1		iskgen2 21351	. 2 |- ( J e. ran 𝑘Gen <-> ( J e. Top /\ (𝑘Gen ` J ) C_ J ) )
2		iskgen3.1	. . . . . . . . . 10 |- X = U. J
3	2	toptopon 20722	. . . . . . . . 9 |- ( J e. Top <-> J e. (TopOn ` X ) )
4		elkgen 21339	. . . . . . . . 9 |- ( J e. (TopOn ` X ) -> ( x e. (𝑘Gen ` J ) <-> ( x C_ X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) ) )
5	3, 4	sylbi 207	. . . . . . . 8 |- ( J e. Top -> ( x e. (𝑘Gen ` J ) <-> ( x C_ X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) ) )
6		vex 3203	. . . . . . . . . 10 |- x e. _V
7	6	elpw 4164	. . . . . . . . 9 |- ( x e. ~P X <-> x C_ X )
8	7	anbi1i 731	. . . . . . . 8 |- ( ( x e. ~P X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) <-> ( x C_ X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) )
9	5, 8	syl6bbr 278	. . . . . . 7 |- ( J e. Top -> ( x e. (𝑘Gen ` J ) <-> ( x e. ~P X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) ) )
10	9	imbi1d 331	. . . . . 6 |- ( J e. Top -> ( ( x e. (𝑘Gen ` J ) -> x e. J ) <-> ( ( x e. ~P X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) -> x e. J ) ) )
11		impexp 462	. . . . . 6 |- ( ( ( x e. ~P X /\ A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) ) -> x e. J ) <-> ( x e. ~P X -> ( A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) -> x e. J ) ) )
12	10, 11	syl6bb 276	. . . . 5 |- ( J e. Top -> ( ( x e. (𝑘Gen ` J ) -> x e. J ) <-> ( x e. ~P X -> ( A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) -> x e. J ) ) ) )
13	12	albidv 1849	. . . 4 |- ( J e. Top -> ( A. x ( x e. (𝑘Gen ` J ) -> x e. J ) <-> A. x ( x e. ~P X -> ( A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) -> x e. J ) ) ) )
14		dfss2 3591	. . . 4 |- ( (𝑘Gen ` J ) C_ J <-> A. x ( x e. (𝑘Gen ` J ) -> x e. J ) )
15		df-ral 2917	. . . 4 |- ( A. x e. ~P X ( A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) -> x e. J ) <-> A. x ( x e. ~P X -> ( A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) -> x e. J ) ) )
16	13, 14, 15	3bitr4g 303	. . 3 |- ( J e. Top -> ( (𝑘Gen ` J ) C_ J <-> A. x e. ~P X ( A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) -> x e. J ) ) )
17	16	pm5.32i 669	. 2 |- ( ( J e. Top /\ (𝑘Gen ` J ) C_ J ) <-> ( J e. Top /\ A. x e. ~P X ( A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) -> x e. J ) ) )
18	1, 17	bitri 264	1 |- ( J e. ran 𝑘Gen <-> ( J e. Top /\ A. x e. ~P X ( A. k e. ~P X ( ( J ↾t k ) e. Comp -> ( x i^i k ) e. ( J ↾t k ) ) -> x e. J ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ran crn 5115   ` 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-rep 4771  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-3or 1038  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-reu 2919  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-kgen 21337

This theorem is referenced by: (None)