Theorem isperf 20955

Description: Definition of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
lpfval.1	|- X = U. J

Assertion

Ref	Expression
isperf	|- ( J e. Perf <-> ( J e. Top /\ ( ( limPt ` J ) ` X ) = X ) )

Proof of Theorem isperf

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( j = J -> ( limPt ` j ) = ( limPt ` J ) )
2		unieq 4444	. . . . 5 |- ( j = J -> U. j = U. J )
3		lpfval.1	. . . . 5 |- X = U. J
4	2, 3	syl6eqr 2674	. . . 4 |- ( j = J -> U. j = X )
5	1, 4	fveq12d 6197	. . 3 |- ( j = J -> ( ( limPt ` j ) ` U. j ) = ( ( limPt ` J ) ` X ) )
6	5, 4	eqeq12d 2637	. 2 |- ( j = J -> ( ( ( limPt ` j ) ` U. j ) = U. j <-> ( ( limPt ` J ) ` X ) = X ) )
7		df-perf 20941	. 2 |- Perf = { j e. Top | ( ( limPt ` j ) ` U. j ) = U. j }
8	6, 7	elrab2 3366	1 |- ( J e. Perf <-> ( J e. Top /\ ( ( limPt ` J ) ` X ) = X ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   U.cuni 4436   ` cfv 5888   Topctop 20698   limPtclp 20938  Perfcperf 20939

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-perf 20941

This theorem is referenced by:  isperf2  20956  perflp  20958  perftop  20960  restperf  20988