Theorem isbasisg 7611

Description: Express the predicate "B is a basis for a topology."

Assertion

Ref	Expression
isbasisg	|- (B e. C -> (B e. Bases <-> A.x e. B A.y e. B (x i^i y) (_ U.(B i^i P~(x i^i y))))

Distinct variable group:   x,y,B

Proof of Theorem isbasisg

Step	Hyp	Ref	Expression
1		ineq1 2210	. . . . . 6 |- (z = B -> (z i^i P~(x i^i y)) = (B i^i P~(x i^i y)))
2	1	unieqd 2512	. . . . 5 |- (z = B -> U.(z i^i P~(x i^i y)) = U.(B i^i P~(x i^i y)))
3	2	sseq2d 2089	. . . 4 |- (z = B -> ((x i^i y) (_ U.(z i^i P~(x i^i y)) <-> (x i^i y) (_ U.(B i^i P~(x i^i y))))
4	3	raleqd 1791	. . 3 |- (z = B -> (A.y e. z (x i^i y) (_ U.(z i^i P~(x i^i y)) <-> A.y e. B (x i^i y) (_ U.(B i^i P~(x i^i y))))
5	4	raleqd 1791	. 2 |- (z = B -> (A.x e. z A.y e. z (x i^i y) (_ U.(z i^i P~(x i^i y)) <-> A.x e. B A.y e. B (x i^i y) (_ U.(B i^i P~(x i^i y))))
6		df-bases 7594	. 2 |- Bases = {z | A.x e. z A.y e. z (x i^i y) (_ U.(z i^i P~(x i^i y))}
7	5, 6	elab2g 1900	1 |- (B e. C -> (B e. Bases <-> A.x e. B A.y e. B (x i^i y) (_ U.(B i^i P~(x i^i y))))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  A.wral 1645   i^i cin 2046   (_ wss 2047  P~cpw 2401  U.cuni 2503  Basesctb 7590

This theorem is referenced by:  isbasis2g 7612  basis1t 7614

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-v 1812  df-in 2051  df-ss 2053  df-uni 2504  df-bases 7594

Copyright terms: Public domain