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| Mirrors > Home > MPE Home > Th. List > Mathboxes > xrninxp2 | Structured version Visualization version GIF version | ||
| Description: Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020.) |
| Ref | Expression |
|---|---|
| xrninxp2 | ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {〈𝑢, 𝑥〉 ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | inxp2 38368 | . 2 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {〈𝑢, 𝑥〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)} | |
| 2 | an21 644 | . . 3 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))) | |
| 3 | 2 | opabbii 5210 | . 2 ⊢ {〈𝑢, 𝑥〉 ∣ ((𝑢 ∈ 𝐴 ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)} = {〈𝑢, 𝑥〉 ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))} |
| 4 | 1, 3 | eqtri 2765 | 1 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {〈𝑢, 𝑥〉 ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 class class class wbr 5143 {copab 5205 × cxp 5683 ⋉ cxrn 38181 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: inxpxrn 38396 |
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