Theorem finlocfin 21323

Description: A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)

Hypotheses

Ref	Expression
finlocfin.1	|- X = U. J
finlocfin.2	|- Y = U. A

Assertion

Ref	Expression
finlocfin	|- ( ( J e. Top /\ A e. Fin /\ X = Y ) -> A e. ( LocFin ` J ) )

Proof of Theorem finlocfin

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

Step	Hyp	Ref	Expression
1		simp1 1061	. 2 |- ( ( J e. Top /\ A e. Fin /\ X = Y ) -> J e. Top )
2		simp3 1063	. 2 |- ( ( J e. Top /\ A e. Fin /\ X = Y ) -> X = Y )
3		simpl1 1064	. . . . 5 |- ( ( ( J e. Top /\ A e. Fin /\ X = Y ) /\ x e. X ) -> J e. Top )
4		finlocfin.1	. . . . . 6 |- X = U. J
5	4	topopn 20711	. . . . 5 |- ( J e. Top -> X e. J )
6	3, 5	syl 17	. . . 4 |- ( ( ( J e. Top /\ A e. Fin /\ X = Y ) /\ x e. X ) -> X e. J )
7		simpr 477	. . . 4 |- ( ( ( J e. Top /\ A e. Fin /\ X = Y ) /\ x e. X ) -> x e. X )
8		simpl2 1065	. . . . 5 |- ( ( ( J e. Top /\ A e. Fin /\ X = Y ) /\ x e. X ) -> A e. Fin )
9		ssrab2 3687	. . . . 5 |- { s e. A | ( s i^i X ) =/= (/) } C_ A
10		ssfi 8180	. . . . 5 |- ( ( A e. Fin /\ { s e. A | ( s i^i X ) =/= (/) } C_ A ) -> { s e. A | ( s i^i X ) =/= (/) } e. Fin )
11	8, 9, 10	sylancl 694	. . . 4 |- ( ( ( J e. Top /\ A e. Fin /\ X = Y ) /\ x e. X ) -> { s e. A | ( s i^i X ) =/= (/) } e. Fin )
12		eleq2 2690	. . . . . 6 |- ( n = X -> ( x e. n <-> x e. X ) )
13		ineq2 3808	. . . . . . . . 9 |- ( n = X -> ( s i^i n ) = ( s i^i X ) )
14	13	neeq1d 2853	. . . . . . . 8 |- ( n = X -> ( ( s i^i n ) =/= (/) <-> ( s i^i X ) =/= (/) ) )
15	14	rabbidv 3189	. . . . . . 7 |- ( n = X -> { s e. A | ( s i^i n ) =/= (/) } = { s e. A | ( s i^i X ) =/= (/) } )
16	15	eleq1d 2686	. . . . . 6 |- ( n = X -> ( { s e. A | ( s i^i n ) =/= (/) } e. Fin <-> { s e. A | ( s i^i X ) =/= (/) } e. Fin ) )
17	12, 16	anbi12d 747	. . . . 5 |- ( n = X -> ( ( x e. n /\ { s e. A | ( s i^i n ) =/= (/) } e. Fin ) <-> ( x e. X /\ { s e. A | ( s i^i X ) =/= (/) } e. Fin ) ) )
18	17	rspcev 3309	. . . 4 |- ( ( X e. J /\ ( x e. X /\ { s e. A | ( s i^i X ) =/= (/) } e. Fin ) ) -> E. n e. J ( x e. n /\ { s e. A | ( s i^i n ) =/= (/) } e. Fin ) )
19	6, 7, 11, 18	syl12anc 1324	. . 3 |- ( ( ( J e. Top /\ A e. Fin /\ X = Y ) /\ x e. X ) -> E. n e. J ( x e. n /\ { s e. A | ( s i^i n ) =/= (/) } e. Fin ) )
20	19	ralrimiva 2966	. 2 |- ( ( J e. Top /\ A e. Fin /\ X = Y ) -> A. x e. X E. n e. J ( x e. n /\ { s e. A | ( s i^i n ) =/= (/) } e. Fin ) )
21		finlocfin.2	. . 3 |- Y = U. A
22	4, 21	islocfin 21320	. 2 |- ( A e. ( LocFin ` J ) <-> ( J e. Top /\ X = Y /\ A. x e. X E. n e. J ( x e. n /\ { s e. A | ( s i^i n ) =/= (/) } e. Fin ) ) )
23	1, 2, 20, 22	syl3anbrc 1246	1 |- ( ( J e. Top /\ A e. Fin /\ X = Y ) -> A e. ( LocFin ` J ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = 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   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-locfin 21310

This theorem is referenced by:  locfincmp  21329  cmppcmp  29925