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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 5420 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
2 | iunid 4808 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
3 | 2 | xpeq1i 5381 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵) |
4 | 1, 3 | eqtr3i 2804 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 {csn 4398 ∪ ciun 4753 × cxp 5353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-iun 4755 df-opab 4949 df-xp 5361 |
This theorem is referenced by: ralxp 5509 rexxp 5510 mpt2mpt 7029 mpt2mpts 7514 fmpt2 7517 fsumxp 14908 fprodxp 15115 dvfval 24098 indval2 30674 filnetlem3 32963 sge0xp 41570 xpiun 42781 |
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