Theorem iscnrm 23641

Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.)

Hypothesis

Ref	Expression
ist0.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
iscnrm	⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem iscnrm

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4878	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
2		ist0.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	eqtr4di 2814	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
4	3	pweqd 4574	. . 3 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
5		oveq1 7427	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑥) = (𝐽 ↾t 𝑥))
6	5	eleq1d 2846	. . 3 ⊢ (𝑗 = 𝐽 → ((𝑗 ↾t 𝑥) ∈ Nrm ↔ (𝐽 ↾t 𝑥) ∈ Nrm))
7	4, 6	raleqbidv 3335	. 2 ⊢ (𝑗 = 𝐽 → (∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm))
8		df-cnrm 23636	. 2 ⊢ CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm}
9	7, 8	elrab2 3649	1 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  𝒫 cpw 4557  ∪ cuni 4867  (class class class)co 7420   ↾_t crest 17591  Topctop 23211  Nrmcnrm 23628  CNrmccnrm 23629

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423  df-cnrm 23636

This theorem is used by:  cnrmtop  23655  iscnrm2  23656  cnrmi  23678  iscnrm3  50059