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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 5314 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
2 | iunid 4709 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
3 | 2 | xpeq1i 5275 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵) |
4 | 1, 3 | eqtr3i 2795 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 {csn 4316 ∪ ciun 4654 × cxp 5247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-iun 4656 df-opab 4847 df-xp 5255 |
This theorem is referenced by: ralxp 5402 rexxp 5403 mpt2mpt 6899 mpt2mpts 7384 fmpt2 7387 fsumxp 14711 fprodxp 14919 dvfval 23881 indval2 30416 filnetlem3 32712 sge0xp 41163 xpiun 42294 |
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