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Theorem ishaus 21865
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 4845 . . . 4 (𝑗 = 𝐽 𝑗 = 𝐽)
2 ist0.1 . . . 4 𝑋 = 𝐽
31, 2syl6eqr 2879 . . 3 (𝑗 = 𝐽 𝑗 = 𝑋)
4 rexeq 3412 . . . . . 6 (𝑗 = 𝐽 → (∃𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
54rexeqbi1dv 3410 . . . . 5 (𝑗 = 𝐽 → (∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅) ↔ ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)))
65imbi2d 342 . . . 4 (𝑗 = 𝐽 → ((𝑥𝑦 → ∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
73, 6raleqbidv 3407 . . 3 (𝑗 = 𝐽 → (∀𝑦 𝑗(𝑥𝑦 → ∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ ∀𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
83, 7raleqbidv 3407 . 2 (𝑗 = 𝐽 → (∀𝑥 𝑗𝑦 𝑗(𝑥𝑦 → ∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅)) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
9 df-haus 21858 . 2 Haus = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(𝑥𝑦 → ∃𝑛𝑗𝑚𝑗 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))}
108, 9elrab2 3687 1 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑦 → ∃𝑛𝐽𝑚𝐽 (𝑥𝑛𝑦𝑚 ∧ (𝑛𝑚) = ∅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  cin 3939  c0 4295   cuni 4837  Topctop 21436  Hauscha 21851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-uni 4838  df-haus 21858
This theorem is referenced by:  hausnei  21871  haustop  21874  ishaus2  21894  cnhaus  21897  dishaus  21925  pthaus  22181  hausdiag  22188  txhaus  22190  xkohaus  22196
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