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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 5441 | . . 3 ⊢ Rel (𝑅 ∩ (𝐴 × 𝐵)) | |
2 | dfrel4v 5801 | . . 3 ⊢ (Rel (𝑅 ∩ (𝐴 × 𝐵)) ↔ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦}) | |
3 | 1, 2 | mpbi 222 | . 2 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦} |
4 | brinxp2 5383 | . . 3 ⊢ (𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)) | |
5 | 4 | opabbii 4910 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ 𝑥(𝑅 ∩ (𝐴 × 𝐵))𝑦} = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)} |
6 | 3, 5 | eqtri 2821 | 1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥𝑅𝑦)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∩ cin 3768 class class class wbr 4843 {copab 4905 × cxp 5310 Rel wrel 5317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-rel 5319 df-cnv 5320 |
This theorem is referenced by: xrninxp 34644 xrninxp2 34645 |
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