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Mirrors > Home > MPE Home > Th. List > resiun1 | Structured version Visualization version GIF version |
Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) (Proof shortened by JJ, 25-Aug-2021.) |
Ref | Expression |
---|---|
resiun1 | ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunin1 4985 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ (𝐶 × V)) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) | |
2 | df-res 5560 | . . . 4 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))) |
4 | 3 | iuneq2i 4931 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ (𝐶 × V)) |
5 | df-res 5560 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) | |
6 | 1, 4, 5 | 3eqtr4ri 2852 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ∪ ciun 4910 × cxp 5546 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 df-iun 4912 df-res 5560 |
This theorem is referenced by: (None) |
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