Theorem sscmp 21208

Description: A subset of a compact topology (i.e. a coarser topology) is compact. (Contributed by Mario Carneiro, 20-Mar-2015.)

Hypothesis

Ref	Expression
sscmp.1	|- X = U. K

Assertion

Ref	Expression
sscmp	|- ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) -> J e. Comp )

Proof of Theorem sscmp

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 20718	. . 3 |- ( J e. (TopOn ` X ) -> J e. Top )
2	1	3ad2ant1 1082	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) -> J e. Top )
3		elpwi 4168	. . . 4 |- ( x e. ~P J -> x C_ J )
4		simpl2 1065	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> K e. Comp )
5		simprl 794	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> x C_ J )
6		simpl3 1066	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> J C_ K )
7	5, 6	sstrd 3613	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> x C_ K )
8		simpl1 1064	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> J e. (TopOn ` X ) )
9		toponuni 20719	. . . . . . . . 9 |- ( J e. (TopOn ` X ) -> X = U. J )
10	8, 9	syl 17	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> X = U. J )
11		simprr 796	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> U. J = U. x )
12	10, 11	eqtrd 2656	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> X = U. x )
13		sscmp.1	. . . . . . . 8 |- X = U. K
14	13	cmpcov 21192	. . . . . . 7 |- ( ( K e. Comp /\ x C_ K /\ X = U. x ) -> E. y e. ( ~P x i^i Fin ) X = U. y )
15	4, 7, 12, 14	syl3anc 1326	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> E. y e. ( ~P x i^i Fin ) X = U. y )
16	10	eqeq1d 2624	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> ( X = U. y <-> U. J = U. y ) )
17	16	rexbidv 3052	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> ( E. y e. ( ~P x i^i Fin ) X = U. y <-> E. y e. ( ~P x i^i Fin ) U. J = U. y ) )
18	15, 17	mpbid 222	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ ( x C_ J /\ U. J = U. x ) ) -> E. y e. ( ~P x i^i Fin ) U. J = U. y )
19	18	expr 643	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ x C_ J ) -> ( U. J = U. x -> E. y e. ( ~P x i^i Fin ) U. J = U. y ) )
20	3, 19	sylan2 491	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) /\ x e. ~P J ) -> ( U. J = U. x -> E. y e. ( ~P x i^i Fin ) U. J = U. y ) )
21	20	ralrimiva 2966	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) -> A. x e. ~P J ( U. J = U. x -> E. y e. ( ~P x i^i Fin ) U. J = U. y ) )
22		eqid 2622	. . 3 |- U. J = U. J
23	22	iscmp 21191	. 2 |- ( J e. Comp <-> ( J e. Top /\ A. x e. ~P J ( U. J = U. x -> E. y e. ( ~P x i^i Fin ) U. J = U. y ) ) )
24	2, 21, 23	sylanbrc 698	1 |- ( ( J e. (TopOn ` X ) /\ K e. Comp /\ J C_ K ) -> J e. Comp )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ` cfv 5888   Fincfn 7955   Topctop 20698  TopOnctopon 20715   Compccmp 21189

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-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-topon 20716  df-cmp 21190

This theorem is referenced by:  kgencmp2  21349  kgen2ss  21358