Theorem iscnrm 23696

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 4883	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
2		ist0.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	eqtr4di 2816	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
4	3	pweqd 4577	. . 3 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
5		oveq1 7458	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑥) = (𝐽 ↾t 𝑥))
6	5	eleq1d 2849	. . 3 ⊢ (𝑗 = 𝐽 → ((𝑗 ↾t 𝑥) ∈ Nrm ↔ (𝐽 ↾t 𝑥) ∈ Nrm))
7	4, 6	raleqbidv 3338	. 2 ⊢ (𝑗 = 𝐽 → (∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm))
8		df-cnrm 23691	. 2 ⊢ CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm}
9	7, 8	elrab2 3652	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 2147  ∀wral 3080  𝒫 cpw 4560  ∪ cuni 4872  (class class class)co 7451   ↾_t crest 17641  Topctop 23266  Nrmcnrm 23683  CNrmccnrm 23684

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-ov 7454  df-cnrm 23691

This theorem is used by:  cnrmtop  23710  iscnrm2  23711  cnrmi  23733  iscnrm3  50397