Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ist1 Structured version   Visualization version   GIF version

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 ∧ ∀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 4380 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
2 ist0.1 . . . . . 6 𝑋 = 𝐽
31, 2syl6eqr 2662 . . . . 5 (𝑥 = 𝐽 𝑥 = 𝑋)
43eleq2d 2673 . . . 4 (𝑥 = 𝐽 → (𝑎 𝑥𝑎𝑋))
5 fveq2 6103 . . . . 5 (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽))
65eleq2d 2673 . . . 4 (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽)))
74, 6imbi12d 333 . . 3 (𝑥 = 𝐽 → ((𝑎 𝑥 → {𝑎} ∈ (Clsd‘𝑥)) ↔ (𝑎𝑋 → {𝑎} ∈ (Clsd‘𝐽))))
87ralbidv2 2967 . 2 (𝑥 = 𝐽 → (∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))
9 df-t1 20928 . 2 Fre = {𝑥 ∈ Top ∣ ∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥)}
108, 9elrab2 3333 1 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))
 Colors of variables: wff setvar class 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
 Copyright terms: Public domain W3C validator