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Theorem ist1 21044
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 4415 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
2 ist0.1 . . . . . 6 𝑋 = 𝐽
31, 2syl6eqr 2673 . . . . 5 (𝑥 = 𝐽 𝑥 = 𝑋)
43eleq2d 2684 . . . 4 (𝑥 = 𝐽 → (𝑎 𝑥𝑎𝑋))
5 fveq2 6153 . . . . 5 (𝑥 = 𝐽 → (Clsd‘𝑥) = (Clsd‘𝐽))
65eleq2d 2684 . . . 4 (𝑥 = 𝐽 → ({𝑎} ∈ (Clsd‘𝑥) ↔ {𝑎} ∈ (Clsd‘𝐽)))
74, 6imbi12d 334 . . 3 (𝑥 = 𝐽 → ((𝑎 𝑥 → {𝑎} ∈ (Clsd‘𝑥)) ↔ (𝑎𝑋 → {𝑎} ∈ (Clsd‘𝐽))))
87ralbidv2 2979 . 2 (𝑥 = 𝐽 → (∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥) ↔ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))
9 df-t1 21037 . 2 Fre = {𝑥 ∈ Top ∣ ∀𝑎 𝑥{𝑎} ∈ (Clsd‘𝑥)}
108, 9elrab2 3352 1 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋 {𝑎} ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {csn 4153   cuni 4407  cfv 5852  Topctop 20626  Clsdccld 20739  Frect1 21030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-t1 21037
This theorem is referenced by:  t1sncld  21049  t1ficld  21050  t1top  21053  ist1-2  21070  cnt1  21073  ordtt1  21102  qtopt1  29702  onint1  32117
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