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Mirrors > Home > MPE Home > Th. List > iunxpconst | Structured version Visualization version GIF version |
Description: Membership in a union of Cartesian products when the second factor is constant. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
iunxpconst | ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpiundir 5760 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
2 | iunid 5065 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
3 | 2 | xpeq1i 5715 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵) |
4 | 1, 3 | eqtr3i 2765 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 {csn 4631 ∪ ciun 4996 × 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: ralxp 5855 rexxp 5856 mpompt 7547 mpompts 8089 fmpo 8092 fsumxp 15805 fprodxp 16015 dvfval 25947 indval2 33995 filnetlem3 36363 sge0xp 46385 xpiun 48002 |
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