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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 43534 | 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 1535 ∃wex 1774 {cab 2710 {crab 3432 ∩ cin 3962 𝒫 cpw 4605 ∩ cint 4954 × cxp 5682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5301 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-tru 1538 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-in 3970 df-ss 3980 df-pw 4607 df-int 4955 |
This theorem is referenced by: relintab 43538 |
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