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Mirrors > Home > MPE Home > Th. List > Mathboxes > inxp2 | Structured version Visualization version GIF version |
Description: Intersection with a Cartesian product. (Contributed by Peter Mazsa, 18-Jul-2019.) |
Ref | Expression |
---|---|
inxp2 | ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relinxp 34412 | . . 3 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵)) | |
2 | dfrel4v 5724 | . . 3 ⊢ (Rel (𝑅 ∩ (𝐴 × 𝐵)) ↔ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦}) | |
3 | 1, 2 | mpbi 220 | . 2 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦} |
4 | brinxp2ALTV 34377 | . . 3 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)) | |
5 | 4 | opabbii 4852 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)} |
6 | 3, 5 | eqtri 2793 | 1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∩ cin 3722 class class class wbr 4787 {copab 4847 × cxp 5248 Rel wrel 5255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-br 4788 df-opab 4848 df-xp 5256 df-rel 5257 df-cnv 5258 |
This theorem is referenced by: xrninxp 34492 xrninxp2 34493 |
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