Theorem isbasisg 12054

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 3236	. . . . . 6 |- ( z = B -> ( z i^i ~P ( x i^i y ) ) = ( B i^i ~P ( x i^i y ) ) )
2	1	unieqd 3713	. . . . 5 |- ( z = B -> U. ( z i^i ~P ( x i^i y ) ) = U. ( B i^i ~P ( x i^i y ) ) )
3	2	sseq2d 3093	. . . 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 2608	. . 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 2608	. 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 12053	. 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 2800	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 1314   e. wcel 1463   A.wral 2390   i^i cin 3036   C_ wss 3037   ~Pcpw 3476   U.cuni 3702   TopBasesctb 12052

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-in 3043  df-ss 3050  df-uni 3703  df-bases 12053

This theorem is referenced by:  isbasis2g  12055  basis1  12057  baspartn  12060