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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 48001 | . . . . 5 ⊢ ({𝑥} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} | |
2 | 1 | eqcomi 2744 | . . . 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 5018 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} = ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶) |
5 | iunxpconst 5761 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶) = (𝐵 × 𝐶) | |
6 | 4, 5 | eqtr2i 2764 | 1 ⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 {csn 4631 ∪ ciun 4996 {copab 5210 × cxp 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-iun 4998 df-opab 5211 df-xp 5695 |
This theorem is referenced by: (None) |
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