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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 22387, unimax 4910, 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 3923 | . 2 ⊢ (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵} | |
2 | elpw2g 5306 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)) | |
3 | 2 | rabbidv 3418 | . 2 ⊢ (𝐵 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝒫 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵}) |
4 | 1, 3 | eqtrid 2789 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3410 ∩ cin 3914 ⊆ wss 3915 𝒫 cpw 4565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3411 df-v 3450 df-in 3922 df-ss 3932 df-pw 4567 |
This theorem is referenced by: toplatglb 47100 |
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