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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 5687 | . . 3 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵)) | |
2 | dfrel4v 6047 | . . 3 ⊢ (Rel (𝑅 ∩ (𝐴 × 𝐵)) ↔ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦}) | |
3 | 1, 2 | mpbi 232 | . 2 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦} |
4 | brinxp2 5629 | . . 3 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)) | |
5 | 4 | opabbii 5133 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)} |
6 | 3, 5 | eqtri 2844 | 1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 class class class wbr 5066 {copab 5128 × cxp 5553 Rel wrel 5560 |
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 ax-nul 5210 ax-pr 5330 |
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-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 |
This theorem is referenced by: xrninxp 35655 xrninxp2 35656 |
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