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| Mirrors > Home > MPE Home > Th. List > inxpOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of inxp 5842 as of 5-May-2025. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| inxpOLD | ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | inopab 5839 | . . 3 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∩ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}) = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))} | |
| 2 | an4 656 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))) | |
| 3 | elin 3967 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶)) | |
| 4 | elin 3967 | . . . . . 6 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐷) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷)) | |
| 5 | 3, 4 | anbi12i 628 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐶) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐷))) | 
| 6 | 2, 5 | bitr4i 278 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) ↔ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))) | 
| 7 | 6 | opabbii 5210 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))} | 
| 8 | 1, 7 | eqtri 2765 | . 2 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∩ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))} | 
| 9 | df-xp 5691 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} | |
| 10 | df-xp 5691 | . . 3 ⊢ (𝐶 × 𝐷) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)} | |
| 11 | 9, 10 | ineq12i 4218 | . 2 ⊢ ((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ({〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} ∩ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)}) | 
| 12 | df-xp 5691 | . 2 ⊢ ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷)) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (𝐴 ∩ 𝐶) ∧ 𝑦 ∈ (𝐵 ∩ 𝐷))} | |
| 13 | 8, 11, 12 | 3eqtr4i 2775 | 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 {copab 5205 × 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-10 2141 ax-12 2177 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-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 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: (None) | 
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