Theorem ist1 20935

Description: The predicate 𝐽 is T₁. (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 4380	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
2		ist0.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	syl6eqr 2662	. . . . 5 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
4	3	eleq2d 2673	. . . 4 ⊢ (𝑥 = 𝐽 → (𝑎 ∈ ∪ 𝑥 ↔ 𝑎 ∈ 𝑋))
5		fveq2 6103	. . . . 5 ⊢ (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽))
6	5	eleq2d 2673	. . . 4 ⊢ (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽)))
7	4, 6	imbi12d 333	. . 3 ⊢ (𝑥 = 𝐽 → ((𝑎 ∈ ∪ 𝑥 → {𝑎} ∈ (Clsd‘𝑥)) ↔ (𝑎 ∈ 𝑋 → {𝑎} ∈ (Clsd‘𝐽))))
8	7	ralbidv2 2967	. 2 ⊢ (𝑥 = 𝐽 → (∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))
9		df-t1 20928	. 2 ⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)}
10	8, 9	elrab2 3333	1 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {csn 4125  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630  Frect1 20921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-t1 20928

This theorem is referenced by:  t1sncld  20940  t1ficld  20941  t1top  20944  ist1-2  20961  cnt1  20964  ordtt1  20993  qtopt1  29230  onint1  31618