Theorem isbasisg 13684

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 3331	. . . . . 6 |- ( z = B -> ( z i^i ~P ( x i^i y ) ) = ( B i^i ~P ( x i^i y ) ) )
2	1	unieqd 3822	. . . . 5 |- ( z = B -> U. ( z i^i ~P ( x i^i y ) ) = U. ( B i^i ~P ( x i^i y ) ) )
3	2	sseq2d 3187	. . . 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 2681	. . 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 2681	. 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 13683	. 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 2886	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 105   = wceq 1353   e. wcel 2148   A.wral 2455   i^i cin 3130   C_ wss 3131   ~Pcpw 3577   U.cuni 3811   TopBasesctb 13682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-in 3137  df-ss 3144  df-uni 3812  df-bases 13683

This theorem is referenced by:  isbasis2g  13685  basis1  13687  baspartn  13690