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Theorem ist1 23329
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 ∧ ∀𝑎𝑋 {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.
StepHypRef Expression
1 unieq 4918 . . . 4 (𝑥 = 𝐽 𝑥 = 𝐽)
2 ist0.1 . . . 4 𝑋 = 𝐽
31, 2eqtr4di 2795 . . 3 (𝑥 = 𝐽 𝑥 = 𝑋)
4 fveq2 6906 . . . 4 (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽))
54eleq2d 2827 . . 3 (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽)))
63, 5raleqbidv 3346 . 2 (𝑥 = 𝐽 → (∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))
7 df-t1 23322 . 2 Fre = {𝑥 ∈ Top ∣ ∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥)}
86, 7elrab2 3695 1 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {csn 4626   cuni 4907  cfv 6561  Topctop 22899  Clsdccld 23024  Frect1 23315
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-t1 23322
This theorem is referenced by:  t1sncld  23334  t1ficld  23335  t1top  23338  ist1-2  23355  cnt1  23358  ordtt1  23387  qtopt1  33834  zarmxt1  33879  onint1  36450
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