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| Mirrors > Home > MPE Home > Th. List > Mathboxes > xpiun | Structured version Visualization version GIF version | ||
| Description: A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.) | 
| Ref | Expression | 
|---|---|
| xpiun | ⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | xpsnopab 48078 | . . . . 5 ⊢ ({𝑥} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} | |
| 2 | 1 | eqcomi 2745 | . . . 4 ⊢ {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} = ({𝑥} × 𝐶) | 
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐵 → {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} = ({𝑥} × 𝐶)) | 
| 4 | 3 | iuneq2i 5012 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} = ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶) | 
| 5 | iunxpconst 5757 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶) = (𝐵 × 𝐶) | |
| 6 | 4, 5 | eqtr2i 2765 | 1 ⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2107 {csn 4625 ∪ ciun 4990 {copab 5204 × cxp 5682 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-iun 4992 df-opab 5205 df-xp 5690 | 
| This theorem is referenced by: (None) | 
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