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Theorem isnrm 21049
Description: The predicate "is a normal space." Much like the case for regular spaces, normal does not imply Hausdorff or even regular. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
isnrm (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
Distinct variable group:   𝑥,𝑦,𝑧,𝐽

Proof of Theorem isnrm
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6148 . . . . 5 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
21ineq1d 3791 . . . 4 (𝑗 = 𝐽 → ((Clsd‘𝑗) ∩ 𝒫 𝑥) = ((Clsd‘𝐽) ∩ 𝒫 𝑥))
3 fveq2 6148 . . . . . . . 8 (𝑗 = 𝐽 → (cls‘𝑗) = (cls‘𝐽))
43fveq1d 6150 . . . . . . 7 (𝑗 = 𝐽 → ((cls‘𝑗)‘𝑧) = ((cls‘𝐽)‘𝑧))
54sseq1d 3611 . . . . . 6 (𝑗 = 𝐽 → (((cls‘𝑗)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑧) ⊆ 𝑥))
65anbi2d 739 . . . . 5 (𝑗 = 𝐽 → ((𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
76rexeqbi1dv 3136 . . . 4 (𝑗 = 𝐽 → (∃𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
82, 7raleqbidv 3141 . . 3 (𝑗 = 𝐽 → (∀𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∀𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
98raleqbi1dv 3135 . 2 (𝑗 = 𝐽 → (∀𝑥𝑗𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥) ↔ ∀𝑥𝐽𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
10 df-nrm 21031 . 2 Nrm = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦 ∈ ((Clsd‘𝑗) ∩ 𝒫 𝑥)∃𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
119, 10elrab2 3348 1 (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑥)∃𝑧𝐽 (𝑦𝑧 ∧ ((cls‘𝐽)‘𝑧) ⊆ 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555  𝒫 cpw 4130  cfv 5847  Topctop 20617  Clsdccld 20730  clsccl 20732  Nrmcnrm 21024
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 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-nrm 21031
This theorem is referenced by:  nrmtop  21050  nrmsep3  21069  isnrm2  21072  kqnrmlem1  21456  kqnrmlem2  21457  nrmhmph  21507
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