Theorem iscnrm 21933

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 4851	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
2		ist0.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	syl6eqr 2876	. . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋)
4	3	pweqd 4560	. . 3 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋)
5		oveq1 7165	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑥) = (𝐽 ↾t 𝑥))
6	5	eleq1d 2899	. . 3 ⊢ (𝑗 = 𝐽 → ((𝑗 ↾t 𝑥) ∈ Nrm ↔ (𝐽 ↾t 𝑥) ∈ Nrm))
7	4, 6	raleqbidv 3403	. 2 ⊢ (𝑗 = 𝐽 → (∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm))
8		df-cnrm 21928	. 2 ⊢ CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm}
9	7, 8	elrab2 3685	1 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  𝒫 cpw 4541  ∪ cuni 4840  (class class class)co 7158   ↾_t crest 16696  Topctop 21503  Nrmcnrm 21920  CNrmccnrm 21921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161  df-cnrm 21928

This theorem is referenced by:  cnrmtop  21947  iscnrm2  21948  cnrmi  21970