Theorem isbasisg 7617

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

Assertion

Ref	Expression
isbasisg	⊢ (B ∈ C → (B ∈ Bases ↔ ∀x ∈ B ∀y ∈ B (x ∩ y) ⊆ ∪(B ∩ ℘(x ∩ y))))

Distinct variable group:   x,y,B

Proof of Theorem isbasisg

Step	Hyp	Ref	Expression
1		ineq1 2214	. . . . . 6 ⊢ (z = B → (z ∩ ℘(x ∩ y)) = (B ∩ ℘(x ∩ y)))
2	1	unieqd 2517	. . . . 5 ⊢ (z = B → ∪(z ∩ ℘(x ∩ y)) = ∪(B ∩ ℘(x ∩ y)))
3	2	sseq2d 2093	. . . 4 ⊢ (z = B → ((x ∩ y) ⊆ ∪(z ∩ ℘(x ∩ y)) ↔ (x ∩ y) ⊆ ∪(B ∩ ℘(x ∩ y))))
4	3	raleqd 1794	. . 3 ⊢ (z = B → (∀y ∈ z (x ∩ y) ⊆ ∪(z ∩ ℘(x ∩ y)) ↔ ∀y ∈ B (x ∩ y) ⊆ ∪(B ∩ ℘(x ∩ y))))
5	4	raleqd 1794	. 2 ⊢ (z = B → (∀x ∈ z ∀y ∈ z (x ∩ y) ⊆ ∪(z ∩ ℘(x ∩ y)) ↔ ∀x ∈ B ∀y ∈ B (x ∩ y) ⊆ ∪(B ∩ ℘(x ∩ y))))
6		df-bases 7603	. 2 ⊢ Bases = {z∣∀x ∈ z ∀y ∈ z (x ∩ y) ⊆ ∪(z ∩ ℘(x ∩ y))}
7	5, 6	elab2g 1903	1 ⊢ (B ∈ C → (B ∈ Bases ↔ ∀x ∈ B ∀y ∈ B (x ∩ y) ⊆ ∪(B ∩ ℘(x ∩ y))))

Syntax hints:   → wi 3   ↔ wb 146   = wceq 958   ∈ wcel 960  ∀wral 1648   ∩ cin 2050   ⊆ wss 2051  ℘cpw 2406  ∪cuni 2508  Basesctb 7599

This theorem is referenced by:  isbasis2g 7618  basis1t 7620

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ral 1652  df-v 1815  df-in 2055  df-ss 2057  df-uni 2509  df-bases 7603

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