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Mirrors > Home > MPE Home > Th. List > Mathboxes > inpw | Structured version Visualization version GIF version |
Description: Two ways of expressing a collection of subsets as seen in df-ntr 22515, unimax 4947, and others (Contributed by Zhi Wang, 27-Sep-2024.) |
Ref | Expression |
---|---|
inpw | ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin5 3955 | . 2 ⊢ (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵} | |
2 | elpw2g 5343 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)) | |
3 | 2 | rabbidv 3440 | . 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵}) |
4 | 1, 3 | eqtrid 2784 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {crab 3432 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-in 3954 df-ss 3964 df-pw 4603 |
This theorem is referenced by: toplatglb 47579 |
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