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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xpinintabd | Structured version Visualization version GIF version |
Description: Value of the intersection of Cartesian product with the intersection of a nonempty class. (Contributed by RP, 12-Aug-2020.) |
Ref | Expression |
---|---|
xpinintabd.x | ⊢ (𝜑 → ∃𝑥𝜓) |
Ref | Expression |
---|---|
xpinintabd | ⊢ (𝜑 → ((𝐴 × 𝐵) ∩ ∩ {𝑥 ∣ 𝜓}) = ∩ {𝑤 ∈ 𝒫 (𝐴 × 𝐵) ∣ ∃𝑥(𝑤 = ((𝐴 × 𝐵) ∩ 𝑥) ∧ 𝜓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpinintabd.x | . 2 ⊢ (𝜑 → ∃𝑥𝜓) | |
2 | 1 | inintabd 43151 | 1 ⊢ (𝜑 → ((𝐴 × 𝐵) ∩ ∩ {𝑥 ∣ 𝜓}) = ∩ {𝑤 ∈ 𝒫 (𝐴 × 𝐵) ∣ ∃𝑥(𝑤 = ((𝐴 × 𝐵) ∩ 𝑥) ∧ 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∃wex 1773 {cab 2702 {crab 3418 ∩ cin 3943 𝒫 cpw 4604 ∩ cint 4950 × cxp 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-in 3951 df-ss 3961 df-pw 4606 df-int 4951 |
This theorem is referenced by: relintab 43155 |
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