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Mirrors > Home > MPE Home > Th. List > iscnrm | Structured version Visualization version GIF version |
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 4807 | . . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽) | |
2 | ist0.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 1, 2 | eqtr4di 2791 | . . . 4 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = 𝑋) |
4 | 3 | pweqd 4507 | . . 3 ⊢ (𝑗 = 𝐽 → 𝒫 ∪ 𝑗 = 𝒫 𝑋) |
5 | oveq1 7179 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑥) = (𝐽 ↾t 𝑥)) | |
6 | 5 | eleq1d 2817 | . . 3 ⊢ (𝑗 = 𝐽 → ((𝑗 ↾t 𝑥) ∈ Nrm ↔ (𝐽 ↾t 𝑥) ∈ Nrm)) |
7 | 4, 6 | raleqbidv 3304 | . 2 ⊢ (𝑗 = 𝐽 → (∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm ↔ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm)) |
8 | df-cnrm 22071 | . 2 ⊢ CNrm = {𝑗 ∈ Top ∣ ∀𝑥 ∈ 𝒫 ∪ 𝑗(𝑗 ↾t 𝑥) ∈ Nrm} | |
9 | 7, 8 | elrab2 3591 | 1 ⊢ (𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝑋(𝐽 ↾t 𝑥) ∈ Nrm)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3053 𝒫 cpw 4488 ∪ cuni 4796 (class class class)co 7172 ↾t crest 16799 Topctop 21646 Nrmcnrm 22063 CNrmccnrm 22064 |
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 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rab 3062 df-v 3400 df-un 3848 df-in 3850 df-ss 3860 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-iota 6297 df-fv 6347 df-ov 7175 df-cnrm 22071 |
This theorem is referenced by: cnrmtop 22090 iscnrm2 22091 cnrmi 22113 iscnrm3 45797 |
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