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| Description: The property of being completely or hereditarily normal. (Contributed by Mario Carneiro, 26-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| ist0.1 | ⊢ 𝑋 = ∪ 𝐽 | 
| Ref | Expression | 
|---|---|
| iscnrm | ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | unieq 4918 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
| 2 | ist0.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
| 3 | 1, 2 | eqtr4di 2795 | . . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) | 
| 4 | 3 | pweqd 4617 | . . 3 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋) | 
| 5 | oveq1 7438 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑥) = (𝐽 ↾t 𝑥)) | |
| 6 | 5 | eleq1d 2826 | . . 3 ⊢ (𝑗 = 𝐽 → ((𝑗 ↾t 𝑥) ∈ Nrm ↔ (𝐽 ↾t 𝑥) ∈ Nrm)) | 
| 7 | 4, 6 | raleqbidv 3346 | . 2 ⊢ (𝑗 = 𝐽 → (∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm)) | 
| 8 | df-cnrm 23326 | . 2 ⊢ CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm} | |
| 9 | 7, 8 | elrab2 3695 | 1 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 𝒫 cpw 4600 ∪ cuni 4907 (class class class)co 7431 ↾t crest 17465 Topctop 22899 Nrmcnrm 23318 CNrmccnrm 23319 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-cnrm 23326 | 
| This theorem is referenced by: cnrmtop 23345 iscnrm2 23346 cnrmi 23368 iscnrm3 48849 | 
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