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| 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 5756 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
| 2 | iunid 5059 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
| 3 | 2 | xpeq1i 5710 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵) | 
| 4 | 1, 3 | eqtr3i 2766 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 {csn 4625 ∪ ciun 4990 × 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: ralxp 5851 rexxp 5852 mpompt 7548 mpompts 8091 fmpo 8094 fsumxp 15809 fprodxp 16019 dvfval 25933 indval2 32840 filnetlem3 36382 sge0xp 46449 xpiun 48079 | 
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