Theorem ist1 22709

Description: The predicate "is a T₁ space". (Contributed by FL, 18-Jun-2007.)

Hypothesis

Ref	Expression
ist0.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
ist1	⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))

Distinct variable group:   𝐽,𝑎

Allowed substitution hint:   𝑋(𝑎)

Proof of Theorem ist1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4881	. . . 4 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
2		ist0.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	eqtr4di 2789	. . 3 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
4		fveq2 6847	. . . 4 ⊢ (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽))
5	4	eleq2d 2818	. . 3 ⊢ (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽)))
6	3, 5	raleqbidv 3317	. 2 ⊢ (𝑥 = 𝐽 → (∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))
7		df-t1 22702	. 2 ⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)}
8	6, 7	elrab2 3651	1 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  {csn 4591  ∪ cuni 4870  ‘cfv 6501  Topctop 22279  Clsdccld 22404  Frect1 22695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-t1 22702

This theorem is referenced by:  t1sncld  22714  t1ficld  22715  t1top  22718  ist1-2  22735  cnt1  22738  ordtt1  22767  qtopt1  32505  zarmxt1  32550  onint1  34997