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

Theorem ist1 21901
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 4821 . . . 4 (𝑥 = 𝐽 𝑥 = 𝐽)
2 ist0.1 . . . 4 𝑋 = 𝐽
31, 2syl6eqr 2873 . . 3 (𝑥 = 𝐽 𝑥 = 𝑋)
4 fveq2 6642 . . . 4 (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽))
54eleq2d 2896 . . 3 (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽)))
63, 5raleqbidv 3385 . 2 (𝑥 = 𝐽 → (∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))
7 df-t1 21894 . 2 Fre = {𝑥 ∈ Top ∣ ∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥)}
86, 7elrab2 3659 1 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1537  wcel 2114  wral 3125  {csn 4539   cuni 4810  cfv 6327  Topctop 21473  Clsdccld 21596  Frect1 21887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-iota 6286  df-fv 6335  df-t1 21894
This theorem is referenced by:  t1sncld  21906  t1ficld  21907  t1top  21910  ist1-2  21927  cnt1  21930  ordtt1  21959  qtopt1  31106  onint1  33801
  Copyright terms: Public domain W3C validator