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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 43074 | 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 3419 ∩ cin 3938 𝒫 cpw 4598 ∩ cint 4944 × cxp 5670 |
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 5294 |
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 3052 df-rex 3061 df-rab 3420 df-v 3465 df-in 3946 df-ss 3956 df-pw 4600 df-int 4945 |
This theorem is referenced by: relintab 43078 |
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