Theorem locfinbas 29760

Description: A locally finite cover must cover the base set of its corresponding topological space. (Contributed by Jeff Hankins, 21-Jan-2010.)

Hypotheses

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

Assertion

Ref	Expression
locfinbas	|- ( A e. ( LocFin ` J ) -> X = Y )

Proof of Theorem locfinbas

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

Step	Hyp	Ref	Expression
1		locfinbas.1	. . 3 |- X = U. J
2		locfinbas.2	. . 3 |- Y = U. A
3	1, 2	islocfin 29755	. 2 |- ( A e. ( LocFin ` J ) <-> ( J e. Top /\ X = Y /\ A. s e. X E. n e. J ( s e. n /\ { x e. A | ( x i^i n ) =/= (/) } e. Fin ) ) )
4	3	simp2bi 1007	1 |- ( A e. ( LocFin ` J ) -> X = Y )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1374   e. wcel 1762   =/= wne 2655   A.wral 2807   E.wrex 2808   {crab 2811   i^i cin 3468   (/)c0 3778   U.cuni 4238   ` cfv 5579   Fincfn 7506   Topctop 19154   LocFinclocfin 29721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fv 5587  df-top 19159  df-locfin 29725

This theorem is referenced by:  lfinpfin  29762  locfincmp  29763  locfindis  29764  locfincf  29765