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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 5689 | . . 3 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵)) | |
2 | dfrel4v 6049 | . . 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 5631 | . . 3 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)) | |
5 | 4 | opabbii 5135 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)} |
6 | 3, 5 | eqtri 2846 | 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 3937 class class class wbr 5068 {copab 5130 × cxp 5555 Rel wrel 5562 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 |
This theorem is referenced by: xrninxp 35642 xrninxp2 35643 |
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