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Theorem ist1 21345
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 4580 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
2 ist0.1 . . . . . 6 𝑋 = 𝐽
31, 2syl6eqr 2822 . . . . 5 (𝑥 = 𝐽 𝑥 = 𝑋)
43eleq2d 2835 . . . 4 (𝑥 = 𝐽 → (𝑎 𝑥𝑎𝑋))
5 fveq2 6332 . . . . 5 (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽))
65eleq2d 2835 . . . 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 3132 . 2 (𝑥 = 𝐽 → (∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))
9 df-t1 21338 . 2 Fre = {𝑥 ∈ Top ∣ ∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥)}
108, 9elrab2 3516 1 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060  {csn 4314   cuni 4572  cfv 6031  Topctop 20917  Clsdccld 21040  Frect1 21331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-t1 21338
This theorem is referenced by:  t1sncld  21350  t1ficld  21351  t1top  21354  ist1-2  21371  cnt1  21374  ordtt1  21403  qtopt1  30236  onint1  32779
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