Theorem isbasisg 7561

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 2206	. . . . . 6 |- (z = B -> (z i^i P~(x i^i y)) = (B i^i P~(x i^i y)))
2	1	unieqd 2507	. . . . 5 |- (z = B -> U.(z i^i P~(x i^i y)) = U.(B i^i P~(x i^i y)))
3	2	sseq2d 2085	. . . 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 1788	. . 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 1788	. 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 7544	. 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 1896	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 954   e. wcel 956  A.wral 1642   i^i cin 2042   (_ wss 2043  P~cpw 2397  U.cuni 2498  Basesctb 7540

This theorem is referenced by:  isbasis2g 7562  basis1t 7564

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-v 1808  df-in 2047  df-ss 2049  df-uni 2499  df-bases 7544

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