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Mirrors > Home > MPE Home > Th. List > 2iunin | Structured version Visualization version GIF version |
Description: Rearrange indexed unions over intersection. (Contributed by NM, 18-Dec-2008.) |
Ref | Expression |
---|---|
2iunin | ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunin2 4984 | . . . 4 ⊢ ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷)) |
3 | 2 | iuneq2i 4931 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) |
4 | iunin1 4985 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) | |
5 | 3, 4 | eqtri 2842 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = (∪ 𝑥 ∈ 𝐴 𝐶 ∩ ∪ 𝑦 ∈ 𝐵 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1531 ∈ wcel 2108 ∩ cin 3933 ∪ ciun 4910 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-in 3941 df-ss 3950 df-iun 4912 |
This theorem is referenced by: fpar 7803 |
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