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| Mirrors > Home > MPE Home > Th. List > inxp | Structured version Visualization version GIF version | ||
| Description: Intersection of two Cartesian products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2141, ax-12 2177. (Revised by SN, 5-May-2025.) |
| Ref | Expression |
|---|---|
| inxp | ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | relinxp 5824 | . 2 ⊢ Rel ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) | |
| 2 | relxp 5703 | . 2 ⊢ Rel ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) | |
| 3 | an4 656 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))) | |
| 4 | opelxp 5721 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
| 5 | opelxp 5721 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐶 × 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) | |
| 6 | 4, 5 | anbi12i 628 | . . . 4 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ∧ 〈𝑥, 𝑦〉 ∈ (𝐶 × 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
| 7 | elin 3967 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶)) | |
| 8 | elin 3967 | . . . . 5 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐷) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)) | |
| 9 | 7, 8 | anbi12i 628 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))) |
| 10 | 3, 6, 9 | 3bitr4i 303 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ∧ 〈𝑥, 𝑦〉 ∈ (𝐶 × 𝐷)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))) |
| 11 | elin 3967 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ∧ 〈𝑥, 𝑦〉 ∈ (𝐶 × 𝐷))) | |
| 12 | opelxp 5721 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))) | |
| 13 | 10, 11, 12 | 3bitr4i 303 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) ↔ 〈𝑥, 𝑦〉 ∈ ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))) |
| 14 | 1, 2, 13 | eqrelriiv 5800 | 1 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 〈cop 4632 × cxp 5683 |
| 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-opab 5206 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: xpindi 5844 xpindir 5845 dmxpin 5942 xpssres 6036 xpdisj1 6181 xpdisj2 6182 imainrect 6201 xpima 6202 cnvrescnv 6215 curry1 8129 curry2 8132 fpar 8141 marypha1lem 9473 fpwwe2lem12 10682 hashxplem 14472 sscres 17867 gsumxp 19994 pjfval 21726 pjpm 21728 txbas 23575 txcls 23612 txrest 23639 trust 24238 ressuss 24271 trcfilu 24303 metreslem 24372 ressxms 24538 ressms 24539 mbfmcst 34261 0rrv 34453 poimirlem26 37653 |
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