Theorem cnrmtop 23327

Description: A completely normal space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
cnrmtop	⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top)

Proof of Theorem cnrmtop

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscnrm 23313	. 2 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 ∪ 𝐽(𝐽 ↾t 𝑥) ∈ Nrm))
3	2	simplbi 497	1 ⊢ (𝐽 ∈ CNrm → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2119  ∀wral 3054  𝒫 cpw 4536  ∪ cuni 4845  (class class class)co 7363   ↾_t crest 17381  Topctop 22883  Nrmcnrm 23300  CNrmccnrm 23301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-cnrm 23308

This theorem is referenced by:  restcnrm  23352