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Mirrors > Home > MPE Home > Th. List > iinpw | Structured version Visualization version GIF version |
Description: The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.) |
Ref | Expression |
---|---|
iinpw | ⊢ 𝒫 ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 4428 | . . . 4 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥) | |
2 | selpw 4115 | . . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥) | |
3 | 2 | ralbii 2963 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥) |
4 | 1, 3 | bitr4i 266 | . . 3 ⊢ (𝑦 ⊆ ∩ 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥) |
5 | selpw 4115 | . . 3 ⊢ (𝑦 ∈ 𝒫 ∩ 𝐴 ↔ 𝑦 ⊆ ∩ 𝐴) | |
6 | vex 3176 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | eliin 4461 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥) |
9 | 4, 5, 8 | 3bitr4i 291 | . 2 ⊢ (𝑦 ∈ 𝒫 ∩ 𝐴 ↔ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝒫 𝑥) |
10 | 9 | eqriv 2607 | 1 ⊢ 𝒫 ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝒫 𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ∩ cint 4410 ∩ ciin 4456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 df-int 4411 df-iin 4458 |
This theorem is referenced by: (None) |
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