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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 4982 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 4993 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) | |
2 | incom 4178 | . . . 4 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)) |
4 | 3 | iuneq2i 4940 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) |
5 | incom 4178 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = (𝐵 ∩ ∪ 𝑥 ∈ 𝐴 𝐶) | |
6 | 1, 4, 5 | 3eqtr4i 2854 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ∪ ciun 4919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-iun 4921 |
This theorem is referenced by: 2iunin 4998 resiun1 5873 tgrest 21767 metnrmlem3 23469 limciun 24492 uniin1 30303 disjunsn 30344 measinblem 31479 sstotbnd2 35067 subsaliuncl 42690 sge0iunmptlemre 42746 |
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