Theorem is1stc 21244

Description: The predicate "is a first-countable topology." This can be described as "every point has a countable local basis" - that is, every point has a countable collection of open sets containing it such that every open set containing the point has an open set from this collection as a subset. (Contributed by Jeff Hankins, 22-Aug-2009.)

Hypothesis

Ref	Expression
is1stc.1	|- X = U. J

Assertion

Ref	Expression
is1stc	|- ( J e. 1stc <-> ( J e. Top /\ A. x e. X E. y e. ~P J ( y ~<_ om /\ A. z e. J ( x e. z -> x e. U. ( y i^i ~P z ) ) ) ) )

Distinct variable groups:   x, y, z, J   x, X

Allowed substitution hints:   X( y, z)

Proof of Theorem is1stc

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4444	. . . 4 |- ( j = J -> U. j = U. J )
2		is1stc.1	. . . 4 |- X = U. J
3	1, 2	syl6eqr 2674	. . 3 |- ( j = J -> U. j = X )
4		pweq 4161	. . . 4 |- ( j = J -> ~P j = ~P J )
5		raleq 3138	. . . . 5 |- ( j = J -> ( A. z e. j ( x e. z -> x e. U. ( y i^i ~P z ) ) <-> A. z e. J ( x e. z -> x e. U. ( y i^i ~P z ) ) ) )
6	5	anbi2d 740	. . . 4 |- ( j = J -> ( ( y ~<_ om /\ A. z e. j ( x e. z -> x e. U. ( y i^i ~P z ) ) ) <-> ( y ~<_ om /\ A. z e. J ( x e. z -> x e. U. ( y i^i ~P z ) ) ) ) )
7	4, 6	rexeqbidv 3153	. . 3 |- ( j = J -> ( E. y e. ~P j ( y ~<_ om /\ A. z e. j ( x e. z -> x e. U. ( y i^i ~P z ) ) ) <-> E. y e. ~P J ( y ~<_ om /\ A. z e. J ( x e. z -> x e. U. ( y i^i ~P z ) ) ) ) )
8	3, 7	raleqbidv 3152	. 2 |- ( j = J -> ( A. x e. U. j E. y e. ~P j ( y ~<_ om /\ A. z e. j ( x e. z -> x e. U. ( y i^i ~P z ) ) ) <-> A. x e. X E. y e. ~P J ( y ~<_ om /\ A. z e. J ( x e. z -> x e. U. ( y i^i ~P z ) ) ) ) )
9		df-1stc 21242	. 2 |- 1stc = { j e. Top | A. x e. U. j E. y e. ~P j ( y ~<_ om /\ A. z e. j ( x e. z -> x e. U. ( y i^i ~P z ) ) ) }
10	8, 9	elrab2 3366	1 |- ( J e. 1stc <-> ( J e. Top /\ A. x e. X E. y e. ~P J ( y ~<_ om /\ A. z e. J ( x e. z -> x e. U. ( y i^i ~P z ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   omcom 7065   ~<_ cdom 7953   Topctop 20698   1stcc1stc 21240

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-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-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-1stc 21242

This theorem is referenced by:  is1stc2  21245  1stctop  21246