Theorem ist1 21125

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 4444	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
2		ist0.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	syl6eqr 2674	. . . . 5 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = 𝑋)
4	3	eleq2d 2687	. . . 4 ⊢ (𝑥 = 𝐽 → (𝑎 ∈ ∪ 𝑥 ↔ 𝑎 ∈ 𝑋))
5		fveq2 6191	. . . . 5 ⊢ (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽))
6	5	eleq2d 2687	. . . 4 ⊢ (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽)))
7	4, 6	imbi12d 334	. . 3 ⊢ (𝑥 = 𝐽 → ((𝑎 ∈ ∪ 𝑥 → {𝑎} ∈ (Clsd‘𝑥)) ↔ (𝑎 ∈ 𝑋 → {𝑎} ∈ (Clsd‘𝐽))))
8	7	ralbidv2 2984	. 2 ⊢ (𝑥 = 𝐽 → (∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))
9		df-t1 21118	. 2 ⊢ Fre = {𝑥 ∈ Top ∣ ∀𝑎 ∈ ∪ 𝑥{𝑎} ∈ (Clsd‘𝑥)}
10	8, 9	elrab2 3366	1 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 {𝑎} ∈ (Clsd‘𝐽)))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {csn 4177  ∪ cuni 4436  ‘cfv 5888  Topctop 20698  Clsdccld 20820  Frect1 21111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-t1 21118

This theorem is referenced by:  t1sncld  21130  t1ficld  21131  t1top  21134  ist1-2  21151  cnt1  21154  ordtt1  21183  qtopt1  29902  onint1  32448