| 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 43592 | 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 395 = wceq 1540 ∃wex 1779 {cab 2714 {crab 3436 ∩ cin 3950 𝒫 cpw 4600 ∩ cint 4946 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-pw 4602 df-int 4947 |
| This theorem is referenced by: relintab 43596 |
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