Theorem isnlly 21272

Description: The property of being an n-locally A topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)

Assertion

Ref	Expression
isnlly	|- ( 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 ) )

Distinct variable groups:   x, u, y, A   u, J, x, y

Proof of Theorem isnlly

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . 7 |- ( j = J -> ( nei ` j ) = ( nei ` J ) )
2	1	fveq1d 6193	. . . . . 6 |- ( j = J -> ( ( nei ` j ) ` { y } ) = ( ( nei ` J ) ` { y } ) )
3	2	ineq1d 3813	. . . . 5 |- ( j = J -> ( ( ( nei ` j ) ` { y } ) i^i ~P x ) = ( ( ( nei ` J ) ` { y } ) i^i ~P x ) )
4		oveq1 6657	. . . . . 6 |- ( j = J -> ( j ↾t u ) = ( J ↾t u ) )
5	4	eleq1d 2686	. . . . 5 |- ( j = J -> ( ( j ↾t u ) e. A <-> ( J ↾t u ) e. A ) )
6	3, 5	rexeqbidv 3153	. . . 4 |- ( j = J -> ( E. u e. ( ( ( nei ` j ) ` { y } ) i^i ~P x ) ( j ↾t u ) e. A <-> E. u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) ( J ↾t u ) e. A ) )
7	6	ralbidv 2986	. . 3 |- ( j = J -> ( A. y e. x E. u e. ( ( ( nei ` j ) ` { y } ) i^i ~P x ) ( j ↾t u ) e. A <-> A. y e. x E. u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) ( J ↾t u ) e. A ) )
8	7	raleqbi1dv 3146	. 2 |- ( j = J -> ( A. x e. j A. y e. x E. u e. ( ( ( nei ` j ) ` { y } ) i^i ~P x ) ( j ↾t u ) e. A <-> A. x e. J A. y e. x E. u e. ( ( ( nei ` J ) ` { y } ) i^i ~P x ) ( J ↾t u ) e. A ) )
9		df-nlly 21270	. 2 |- 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 }
10	8, 9	elrab2 3366	1 |- ( 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 ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   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:  nllytop  21276  nllyi  21278  llynlly  21280  nllyss  21283  nllyrest  21289  nllyidm  21292  hausllycmp  21297  cldllycmp  21298  txnlly  21440  cnllycmp  22755