Theorem perftop 19824

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 19819	. 2 |- ( J e. Perf <-> ( J e. Top /\ ( ( limPt ` J ) ` U. J ) = U. J ) )
3	2	simplbi 458	1 |- ( J e. Perf -> J e. Top )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 1823   U.cuni 4235   ` cfv 5570   Topctop 19561   limPtclp 19802  Perfcperf 19803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-perf 19805

This theorem is referenced by:  perfopn  19853