Theorem lfinpfin 21327

Description: A locally finite cover is point-finite. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Assertion

Ref	Expression
lfinpfin	|- ( A e. ( LocFin ` J ) -> A e. PtFin )

Proof of Theorem lfinpfin

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . 8 |- U. J = U. J
2		eqid 2622	. . . . . . . 8 |- U. A = U. A
3	1, 2	locfinbas 21325	. . . . . . 7 |- ( A e. ( LocFin ` J ) -> U. J = U. A )
4	3	eleq2d 2687	. . . . . 6 |- ( A e. ( LocFin ` J ) -> ( x e. U. J <-> x e. U. A ) )
5	4	biimpar 502	. . . . 5 |- ( ( A e. ( LocFin ` J ) /\ x e. U. A ) -> x e. U. J )
6	1	locfinnei 21326	. . . . 5 |- ( ( A e. ( LocFin ` J ) /\ x e. U. J ) -> E. n e. J ( x e. n /\ { s e. A | ( s i^i n ) =/= (/) } e. Fin ) )
7	5, 6	syldan 487	. . . 4 |- ( ( A e. ( LocFin ` J ) /\ x e. U. A ) -> E. n e. J ( x e. n /\ { s e. A | ( s i^i n ) =/= (/) } e. Fin ) )
8		inelcm 4032	. . . . . . . . . 10 |- ( ( x e. s /\ x e. n ) -> ( s i^i n ) =/= (/) )
9	8	expcom 451	. . . . . . . . 9 |- ( x e. n -> ( x e. s -> ( s i^i n ) =/= (/) ) )
10	9	ad2antlr 763	. . . . . . . 8 |- ( ( ( ( A e. ( LocFin ` J ) /\ x e. U. A ) /\ x e. n ) /\ s e. A ) -> ( x e. s -> ( s i^i n ) =/= (/) ) )
11	10	ss2rabdv 3683	. . . . . . 7 |- ( ( ( A e. ( LocFin ` J ) /\ x e. U. A ) /\ x e. n ) -> { s e. A | x e. s } C_ { s e. A | ( s i^i n ) =/= (/) } )
12		ssfi 8180	. . . . . . . 8 |- ( ( { s e. A | ( s i^i n ) =/= (/) } e. Fin /\ { s e. A | x e. s } C_ { s e. A | ( s i^i n ) =/= (/) } ) -> { s e. A | x e. s } e. Fin )
13	12	expcom 451	. . . . . . 7 |- ( { s e. A | x e. s } C_ { s e. A | ( s i^i n ) =/= (/) } -> ( { s e. A | ( s i^i n ) =/= (/) } e. Fin -> { s e. A | x e. s } e. Fin ) )
14	11, 13	syl 17	. . . . . 6 |- ( ( ( A e. ( LocFin ` J ) /\ x e. U. A ) /\ x e. n ) -> ( { s e. A | ( s i^i n ) =/= (/) } e. Fin -> { s e. A | x e. s } e. Fin ) )
15	14	expimpd 629	. . . . 5 |- ( ( A e. ( LocFin ` J ) /\ x e. U. A ) -> ( ( x e. n /\ { s e. A | ( s i^i n ) =/= (/) } e. Fin ) -> { s e. A | x e. s } e. Fin ) )
16	15	rexlimdvw 3034	. . . 4 |- ( ( A e. ( LocFin ` J ) /\ x e. U. A ) -> ( E. n e. J ( x e. n /\ { s e. A | ( s i^i n ) =/= (/) } e. Fin ) -> { s e. A | x e. s } e. Fin ) )
17	7, 16	mpd 15	. . 3 |- ( ( A e. ( LocFin ` J ) /\ x e. U. A ) -> { s e. A | x e. s } e. Fin )
18	17	ralrimiva 2966	. 2 |- ( A e. ( LocFin ` J ) -> A. x e. U. A { s e. A | x e. s } e. Fin )
19	2	isptfin 21319	. 2 |- ( A e. ( LocFin ` J ) -> ( A e. PtFin <-> A. x e. U. A { s e. A | x e. s } e. Fin ) )
20	18, 19	mpbird 247	1 |- ( A e. ( LocFin ` J ) -> A e. PtFin )

Syntax hints:   -> wi 4   /\ wa 384   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   PtFincptfin 21306   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-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-rab 2921  df-v 3202  df-sbc 3436  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-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-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-om 7066  df-er 7742  df-en 7956  df-fin 7959  df-top 20699  df-ptfin 21309  df-locfin 21310

This theorem is referenced by:  locfindis  21333