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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 5174 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | |
2 | iunid 4575 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
3 | 2 | xpeq1i 5135 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 {𝑥} × 𝐵) = (𝐴 × 𝐵) |
4 | 1, 3 | eqtr3i 2646 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 {csn 4177 ∪ ciun 4520 × cxp 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-iun 4522 df-opab 4713 df-xp 5120 |
This theorem is referenced by: ralxp 5263 rexxp 5264 mpt2mpt 6752 mpt2mpts 7234 fmpt2 7237 fsumxp 14503 fprodxp 14712 dvfval 23661 indval2 30076 filnetlem3 32375 sge0xp 40646 xpiun 41766 |
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