Theorem ist1 23546

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 4878	. . . 4 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
2		ist0.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	eqtr4di 2813	. . 3 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
4		fveq2 6878	. . . 4 ⊢ (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽))
5	4	eleq2d 2846	. . 3 ⊢ (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽)))
6	3, 5	raleqbidv 3334	. 2 ⊢ (𝑥 = 𝐽 → (∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))
7		df-t1 23539	. 2 ⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)}
8	6, 7	elrab2 3649	1 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {csn 4584  ∪ cuni 4867  ‘cfv 6533  Topctop 23118  Clsdccld 23241  Frect1 23532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-t1 23539

This theorem is used by:  t1sncld  23551  t1ficld  23552  t1top  23555  ist1-2  23572  cnt1  23575  ordtt1  23604  qtopt1  34345  zarmxt1  34390  onint1  37068