Theorem ist1 23208

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 4882	. . . 4 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
2		ist0.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	eqtr4di 2782	. . 3 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
4		fveq2 6858	. . . 4 ⊢ (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽))
5	4	eleq2d 2814	. . 3 ⊢ (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽)))
6	3, 5	raleqbidv 3319	. 2 ⊢ (𝑥 = 𝐽 → (∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))
7		df-t1 23201	. 2 ⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)}
8	6, 7	elrab2 3662	1 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {csn 4589  ∪ cuni 4871  ‘cfv 6511  Topctop 22780  Clsdccld 22903  Frect1 23194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-t1 23201

This theorem is referenced by:  t1sncld  23213  t1ficld  23214  t1top  23217  ist1-2  23234  cnt1  23237  ordtt1  23266  qtopt1  33825  zarmxt1  33870  onint1  36437