Theorem basis1 13632

Description: Property of a basis. (Contributed by NM, 16-Jul-2006.)

Assertion

Ref	Expression
basis1	|- ( ( B e. TopBases /\ C e. B /\ D e. B ) -> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) )

Proof of Theorem basis1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isbasisg 13629	. . . 4 |- ( B e. TopBases -> ( 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 ) ) ) )
2	1	ibi 176	. . 3 |- ( 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 ) ) )
3		ineq1 3331	. . . . 5 |- ( x = C -> ( x i^i y ) = ( C i^i y ) )
4	3	pweqd 3582	. . . . . . 7 |- ( x = C -> ~P ( x i^i y ) = ~P ( C i^i y ) )
5	4	ineq2d 3338	. . . . . 6 |- ( x = C -> ( B i^i ~P ( x i^i y ) ) = ( B i^i ~P ( C i^i y ) ) )
6	5	unieqd 3822	. . . . 5 |- ( x = C -> U. ( B i^i ~P ( x i^i y ) ) = U. ( B i^i ~P ( C i^i y ) ) )
7	3, 6	sseq12d 3188	. . . 4 |- ( x = C -> ( ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) <-> ( C i^i y ) C_ U. ( B i^i ~P ( C i^i y ) ) ) )
8		ineq2 3332	. . . . 5 |- ( y = D -> ( C i^i y ) = ( C i^i D ) )
9	8	pweqd 3582	. . . . . . 7 |- ( y = D -> ~P ( C i^i y ) = ~P ( C i^i D ) )
10	9	ineq2d 3338	. . . . . 6 |- ( y = D -> ( B i^i ~P ( C i^i y ) ) = ( B i^i ~P ( C i^i D ) ) )
11	10	unieqd 3822	. . . . 5 |- ( y = D -> U. ( B i^i ~P ( C i^i y ) ) = U. ( B i^i ~P ( C i^i D ) ) )
12	8, 11	sseq12d 3188	. . . 4 |- ( y = D -> ( ( C i^i y ) C_ U. ( B i^i ~P ( C i^i y ) ) <-> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) ) )
13	7, 12	rspc2v 2856	. . 3 |- ( ( C e. B /\ D e. B ) -> ( A. x e. B A. y e. B ( x i^i y ) C_ U. ( B i^i ~P ( x i^i y ) ) -> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) ) )
14	2, 13	syl5com 29	. 2 |- ( B e. TopBases -> ( ( C e. B /\ D e. B ) -> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) ) )
15	14	3impib 1201	1 |- ( ( B e. TopBases /\ C e. B /\ D e. B ) -> ( C i^i D ) C_ U. ( B i^i ~P ( C i^i D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   i^i cin 3130   C_ wss 3131   ~Pcpw 3577   U.cuni 3811   TopBasesctb 13627

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-3an 980  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-pw 3579  df-uni 3812  df-bases 13628

This theorem is referenced by: (None)