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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 44026 | . . . . 5 ⊢ ({𝑥} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} | |
2 | 1 | eqcomi 2830 | . . . 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 4932 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} = ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶) |
5 | iunxpconst 5618 | . 2 ⊢ ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶) = (𝐵 × 𝐶) | |
6 | 4, 5 | eqtr2i 2845 | 1 ⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 {csn 4560 ∪ ciun 4911 {copab 5120 × cxp 5547 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-iun 4913 df-opab 5121 df-xp 5555 |
This theorem is referenced by: (None) |
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