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Theorem ishaus 23239
Description: The predicate "is a Hausdorff space". (Contributed by NM, 8-Mar-2007.)
Hypothesis
Ref Expression
ist0.1 𝑋 = 𝐽
Assertion
Ref Expression
ishaus (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
Distinct variable groups:   𝑥,𝑦   𝑚,𝑛,𝑥,𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝑋(𝑚,𝑛)

Proof of Theorem ishaus
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 unieq 4919 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ist0.1 . . . 4 𝑋 = 𝐽
31, 2eqtr4di 2786 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 rexeq 3318 . . . . . 6 (𝑗 = 𝐽 → (∃𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
54rexeqbi1dv 3331 . . . . 5 (𝑗 = 𝐽 → (∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
65imbi2d 340 . . . 4 (𝑗 = 𝐽 → ((𝑥𝑦 → ∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
73, 6raleqbidv 3339 . . 3 (𝑗 = 𝐽 → (∀𝑦 𝑗(𝑥𝑦 → ∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ ∀𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
83, 7raleqbidv 3339 . 2 (𝑗 = 𝐽 → (∀𝑥 𝑗𝑦 𝑗(𝑥𝑦 → ∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
9 df-haus 23232 . 2 Haus = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(𝑥𝑦 → ∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))}
108, 9elrab2 3685 1 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2937  wral 3058  wrex 3067  cin 3946  c0 4323   cuni 4908  Topctop 22808  Hauscha 23225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-in 3954  df-ss 3964  df-uni 4909  df-haus 23232
This theorem is referenced by:  hausnei  23245  haustop  23248  ishaus2  23268  cnhaus  23271  dishaus  23299  pthaus  23555  hausdiag  23562  txhaus  23564  xkohaus  23570
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