Theorem perftop 18902

Description: A perfect space is a topology. (Contributed by Mario Carneiro, 25-Dec-2016.)

Assertion

Ref	Expression
perftop	|- ( J e. Perf -> J e. Top )

Proof of Theorem perftop

Step	Hyp	Ref	Expression
1		eqid 2454	. . 3 |- U. J = U. J
2	1	isperf 18897	. 2 |- ( J e. Perf <-> ( J e. Top /\ ( ( limPt ` J ) ` U. J ) = U. J ) )
3	2	simplbi 460	1 |- ( J e. Perf -> J e. Top )

Syntax hints:   -> wi 4   = wceq 1370   e. wcel 1758   U.cuni 4202   ` cfv 5529   Topctop 18640   limPtclp 18880  Perfcperf 18881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-iota 5492  df-fv 5537  df-perf 18883

This theorem is referenced by:  perfopn  18931