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Mirrors > Home > MPE Home > Th. List > iunin1 | Structured version Visualization version GIF version |
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4988 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
iunin1 | ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunin2 5000 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) | |
2 | incom 4135 | . . . 4 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)) |
4 | 3 | iuneq2i 4946 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) |
5 | incom 4135 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) | |
6 | 1, 4, 5 | 3eqtr4i 2776 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2106 ∩ cin 3886 ∪ ciun 4925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-in 3894 df-ss 3904 df-iun 4927 |
This theorem is referenced by: 2iunin 5005 resiun1 5905 tgrest 22298 metnrmlem3 24012 limciun 25046 uniin1 30877 disjunsn 30919 measinblem 32174 sstotbnd2 35918 subsaliuncl 43856 sge0iunmptlemre 43912 |
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