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Mirrors > Home > ILE Home > Th. List > resiun1 | GIF version |
Description: Distribution of restriction over indexed union. (Contributed by Mario Carneiro, 29-May-2015.) |
Ref | Expression |
---|---|
resiun1 | ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunin2 3936 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐶 × V) ∩ 𝐵) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵) | |
2 | df-res 4623 | . . . . 5 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
3 | incom 3319 | . . . . 5 ⊢ (𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝐵) | |
4 | 2, 3 | eqtri 2191 | . . . 4 ⊢ (𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ 𝐵) |
5 | 4 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ 𝐵)) |
6 | 5 | iuneq2i 3891 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 ((𝐶 × V) ∩ 𝐵) |
7 | df-res 4623 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) | |
8 | incom 3319 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵) | |
9 | 7, 8 | eqtri 2191 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵) |
10 | 1, 6, 9 | 3eqtr4ri 2202 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 Vcvv 2730 ∩ cin 3120 ∪ ciun 3873 × cxp 4609 ↾ cres 4613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-in 3127 df-ss 3134 df-iun 3875 df-res 4623 |
This theorem is referenced by: (None) |
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