Theorem isbasisg 20751

Description: Express the predicate " 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 3807	. . . . . 6 |- ( z = B -> ( z i^i ~P ( x i^i y ) ) = ( B i^i ~P ( x i^i y ) ) )
2	1	unieqd 4446	. . . . 5 |- ( z = B -> U. ( z i^i ~P ( x i^i y ) ) = U. ( B i^i ~P ( x i^i y ) ) )
3	2	sseq2d 3633	. . . 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 3146	. . 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 3146	. 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 20750	. 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 3353	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 196   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   TopBasesctb 20749

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-v 3202  df-in 3581  df-ss 3588  df-uni 4437  df-bases 20750

This theorem is referenced by:  isbasis2g  20752  basis1  20754  basdif0  20757  baspartn  20758  basqtop  21514