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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 5097 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
2 | iunid 4511 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
3 | 2 | xpeq1i 5059 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵) |
4 | 1, 3 | eqtr3i 2634 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 {csn 4125 ∪ ciun 4455 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-iun 4457 df-opab 4644 df-xp 5044 |
This theorem is referenced by: ralxp 5185 rexxp 5186 mpt2mpt 6650 mpt2mpts 7123 fmpt2 7126 fsumxp 14345 fprodxp 14551 dvfval 23467 indval2 29404 filnetlem3 31545 sge0xp 39322 xpiun 41556 |
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