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Mirrors > Home > MPE Home > Th. List > Mathboxes > xpinintabd | Structured version Visualization version GIF version |
Description: Value of the intersection of cross-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 39959 | 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 398 = wceq 1537 ∃wex 1780 {cab 2799 {crab 3142 ∩ cin 3935 𝒫 cpw 4539 ∩ cint 4876 × cxp 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-pw 4541 df-int 4877 |
This theorem is referenced by: relintab 39963 |
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