Theorem nllytop 21276

Description: A locally A space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
nllytop	|- ( J e. 𝑛Locally A -> J e. Top )

Proof of Theorem nllytop

Dummy variables u x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isnlly 21272	. 2 |- ( J e. 𝑛Locally A <-> ( J e. Top /\ A. x e. J A. y e. x E. u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) ( J ↾t u ) e. A ) )
2	1	simplbi 476	1 |- ( J e. 𝑛Locally A -> J e. Top )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   ~Pcpw 4158   {csn 4177   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698   neicnei 20901  𝑛Locally cnlly 21268

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-ral 2917  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-ov 6653  df-nlly 21270

This theorem is referenced by:  nlly2i  21279  restnlly  21285  nllyrest  21289  nllyidm  21292  cldllycmp  21298  llycmpkgen  21355  txnlly  21440  txkgen  21455  xkococnlem  21462  xkococn  21463  cnmptkk  21486  xkofvcn  21487  cnmptk1p  21488  cnmptk2  21489  xkocnv  21617  xkohmeo  21618