Theorem locfincf 21334

Description: A locally finite cover in a coarser topology is locally finite in a finer topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Hypothesis

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

Assertion

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

Proof of Theorem locfincf

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

Step	Hyp	Ref	Expression
1		topontop 20718	. . . . 5 |- ( K e. (TopOn ` X ) -> K e. Top )
2	1	ad2antrr 762	. . . 4 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ x e. ( LocFin ` J ) ) -> K e. Top )
3		toponuni 20719	. . . . . 6 |- ( K e. (TopOn ` X ) -> X = U. K )
4	3	ad2antrr 762	. . . . 5 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ x e. ( LocFin ` J ) ) -> X = U. K )
5		locfincf.1	. . . . . . 7 |- X = U. J
6		eqid 2622	. . . . . . 7 |- U. x = U. x
7	5, 6	locfinbas 21325	. . . . . 6 |- ( x e. ( LocFin ` J ) -> X = U. x )
8	7	adantl 482	. . . . 5 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ x e. ( LocFin ` J ) ) -> X = U. x )
9	4, 8	eqtr3d 2658	. . . 4 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ x e. ( LocFin ` J ) ) -> U. K = U. x )
10	4	eleq2d 2687	. . . . . 6 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ x e. ( LocFin ` J ) ) -> ( y e. X <-> y e. U. K ) )
11	5	locfinnei 21326	. . . . . . . 8 |- ( ( x e. ( LocFin ` J ) /\ y e. X ) -> E. n e. J ( y e. n /\ { s e. x | ( s i^i n ) =/= (/) } e. Fin ) )
12	11	ex 450	. . . . . . 7 |- ( x e. ( LocFin ` J ) -> ( y e. X -> E. n e. J ( y e. n /\ { s e. x | ( s i^i n ) =/= (/) } e. Fin ) ) )
13		ssrexv 3667	. . . . . . . 8 |- ( J C_ K -> ( E. n e. J ( y e. n /\ { s e. x | ( s i^i n ) =/= (/) } e. Fin ) -> E. n e. K ( y e. n /\ { s e. x | ( s i^i n ) =/= (/) } e. Fin ) ) )
14	13	adantl 482	. . . . . . 7 |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( E. n e. J ( y e. n /\ { s e. x | ( s i^i n ) =/= (/) } e. Fin ) -> E. n e. K ( y e. n /\ { s e. x | ( s i^i n ) =/= (/) } e. Fin ) ) )
15	12, 14	sylan9r 690	. . . . . 6 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ x e. ( LocFin ` J ) ) -> ( y e. X -> E. n e. K ( y e. n /\ { s e. x | ( s i^i n ) =/= (/) } e. Fin ) ) )
16	10, 15	sylbird 250	. . . . 5 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ x e. ( LocFin ` J ) ) -> ( y e. U. K -> E. n e. K ( y e. n /\ { s e. x | ( s i^i n ) =/= (/) } e. Fin ) ) )
17	16	ralrimiv 2965	. . . 4 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ x e. ( LocFin ` J ) ) -> A. y e. U. K E. n e. K ( y e. n /\ { s e. x | ( s i^i n ) =/= (/) } e. Fin ) )
18		eqid 2622	. . . . 5 |- U. K = U. K
19	18, 6	islocfin 21320	. . . 4 |- ( x e. ( LocFin ` K ) <-> ( K e. Top /\ U. K = U. x /\ A. y e. U. K E. n e. K ( y e. n /\ { s e. x | ( s i^i n ) =/= (/) } e. Fin ) ) )
20	2, 9, 17, 19	syl3anbrc 1246	. . 3 |- ( ( ( K e. (TopOn ` X ) /\ J C_ K ) /\ x e. ( LocFin ` J ) ) -> x e. ( LocFin ` K ) )
21	20	ex 450	. 2 |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( x e. ( LocFin ` J ) -> x e. ( LocFin ` K ) ) )
22	21	ssrdv 3609	1 |- ( ( K e. (TopOn ` X ) /\ J C_ K ) -> ( LocFin ` J ) C_ ( LocFin ` K ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   ` cfv 5888   Fincfn 7955   Topctop 20698  TopOnctopon 20715   LocFinclocfin 21307

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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-top 20699  df-topon 20716  df-locfin 21310

This theorem is referenced by: (None)