Theorem isbasisg 12836

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 3321	. . . . . 6 |- ( z = B -> ( z i^i ~P ( x i^i y ) ) = ( B i^i ~P ( x i^i y ) ) )
2	1	unieqd 3807	. . . . 5 |- ( z = B -> U. ( z i^i ~P ( x i^i y ) ) = U. ( B i^i ~P ( x i^i y ) ) )
3	2	sseq2d 3177	. . . 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 2673	. . 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 2673	. 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 12835	. 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 2877	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 1348   e. wcel 2141   A.wral 2448   i^i cin 3120   C_ wss 3121   ~Pcpw 3566   U.cuni 3796   TopBasesctb 12834

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-bases 12835

This theorem is referenced by:  isbasis2g  12837  basis1  12839  baspartn  12842