Theorem isbasisg 12200

Description: Express the predicate "the set B is a basis for a topology". (Contributed by NM, 17-Jul-2006.)

Assertion

Ref	Expression
isbasisg	|- ( B e. C -> ( B e. TopBases <-> A. x e. B A. y e. B ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) ) )

Distinct variable group:   x, y, B

Allowed substitution hints:   C( x, y)

Proof of Theorem isbasisg

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 3265	. . . . . 6 |- ( z = B -> ( z i^i ~P ( x i^i y ) ) = ( B i^i ~P ( x i^i y ) ) )
2	1	unieqd 3742	. . . . 5 |- ( z = B -> U. ( z i^i ~P ( x i^i y ) ) = U. ( B i^i ~P ( x i^i y ) ) )
3	2	sseq2d 3122	. . . 4 |- ( z = B -> ( ( x i^i y ) C_ U. ( z i^i ~P ( x i^i y ) ) <-> ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) ) )
4	3	raleqbi1dv 2632	. . 3 |- ( z = B -> ( A. y e. z ( x i^i y ) C_ U. ( z i^i ~P ( x i^i y ) ) <-> A. y e. B ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) ) )
5	4	raleqbi1dv 2632	. 2 |- ( z = B -> ( A. x e. z A. y e. z ( x i^i y ) C_ U. ( z i^i ~P ( x i^i y ) ) <-> A. x e. B A. y e. B ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) ) )
6		df-bases 12199	. 2 |- TopBases = { z | A. x e. z A. y e. z ( x i^i y ) C_ U. ( z i^i ~P ( x i^i y ) ) }
7	5, 6	elab2g 2826	1 |- ( B e. C -> ( B e. TopBases <-> A. x e. B A. y e. B ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   A.wral 2414   i^i cin 3065   C_ wss 3066   ~Pcpw 3505   U.cuni 3731   TopBasesctb 12198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-bases 12199

This theorem is referenced by:  isbasis2g  12201  basis1  12203  baspartn  12206