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Theorem ist1 22578
Description: The predicate "is a T1 space". (Contributed by FL, 18-Jun-2007.)
Hypothesis
Ref Expression
ist0.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
ist1 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ βˆ€π‘Ž ∈ 𝑋 {π‘Ž} ∈ (Clsdβ€˜π½)))
Distinct variable group:   𝐽,π‘Ž
Allowed substitution hint:   𝑋(π‘Ž)

Proof of Theorem ist1
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 unieq 4863 . . . 4 (π‘₯ = 𝐽 β†’ βˆͺ π‘₯ = βˆͺ 𝐽)
2 ist0.1 . . . 4 𝑋 = βˆͺ 𝐽
31, 2eqtr4di 2794 . . 3 (π‘₯ = 𝐽 β†’ βˆͺ π‘₯ = 𝑋)
4 fveq2 6825 . . . 4 (π‘₯ = 𝐽 β†’ (Clsdβ€˜π‘₯) = (Clsdβ€˜π½))
54eleq2d 2822 . . 3 (π‘₯ = 𝐽 β†’ ({π‘Ž} ∈ (Clsdβ€˜π‘₯) ↔ {π‘Ž} ∈ (Clsdβ€˜π½)))
63, 5raleqbidv 3315 . 2 (π‘₯ = 𝐽 β†’ (βˆ€π‘Ž ∈ βˆͺ π‘₯{π‘Ž} ∈ (Clsdβ€˜π‘₯) ↔ βˆ€π‘Ž ∈ 𝑋 {π‘Ž} ∈ (Clsdβ€˜π½)))
7 df-t1 22571 . 2 Fre = {π‘₯ ∈ Top ∣ βˆ€π‘Ž ∈ βˆͺ π‘₯{π‘Ž} ∈ (Clsdβ€˜π‘₯)}
86, 7elrab2 3637 1 (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ βˆ€π‘Ž ∈ 𝑋 {π‘Ž} ∈ (Clsdβ€˜π½)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  βˆ€wral 3061  {csn 4573  βˆͺ cuni 4852  β€˜cfv 6479  Topctop 22148  Clsdccld 22273  Frect1 22564
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-iota 6431  df-fv 6487  df-t1 22571
This theorem is referenced by:  t1sncld  22583  t1ficld  22584  t1top  22587  ist1-2  22604  cnt1  22607  ordtt1  22636  qtopt1  32083  zarmxt1  32128  onint1  34734
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