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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 4509 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 4520 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) | |
2 | incom 3767 | . . . 4 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)) |
4 | 3 | iuneq2i 4475 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) |
5 | incom 3767 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) | |
6 | 1, 4, 5 | 3eqtr4i 2642 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ∪ ciun 4455 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-in 3547 df-ss 3554 df-iun 4457 |
This theorem is referenced by: 2iunin 4524 resiun1 5336 tgrest 20773 metnrmlem3 22472 limciun 23464 uniin1 28750 disjunsn 28789 measinblem 29610 sstotbnd2 32743 subsaliuncl 39252 sge0iunmptlemre 39308 |
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