Theorem locfinbas 20530

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 20525	. 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 1023	1 |- ( A e. ( LocFin ` J ) -> X = Y )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1443   e. wcel 1886   =/= wne 2621   A.wral 2736   E.wrex 2737   {crab 2740   i^i cin 3402   (/)c0 3730   U.cuni 4197   ` cfv 5581   Fincfn 7566   Topctop 19910   LocFinclocfin 20512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fv 5589  df-top 19914  df-locfin 20515

This theorem is referenced by:  lfinpfin  20532  lfinun  20533  locfincmp  20534  locfindis  20538  locfincf  20539