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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 3949 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ((𝐶 × V) ∩ 𝐵) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵) | |
2 | df-res 4637 | . . . . 5 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
3 | incom 3327 | . . . . 5 ⊢ (𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ 𝐵) | |
4 | 2, 3 | eqtri 2198 | . . . 4 ⊢ (𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ 𝐵) |
5 | 4 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ 𝐵)) |
6 | 5 | iuneq2i 3904 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 ((𝐶 × V) ∩ 𝐵) |
7 | df-res 4637 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) | |
8 | incom 3327 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∩ (𝐶 × V)) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵) | |
9 | 7, 8 | eqtri 2198 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ((𝐶 × V) ∩ ∪ 𝑥 ∈ 𝐴 𝐵) |
10 | 1, 6, 9 | 3eqtr4ri 2209 | 1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ↾ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 ↾ 𝐶) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∩ cin 3128 ∪ ciun 3886 × cxp 4623 ↾ cres 4627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-in 3135 df-ss 3142 df-iun 3888 df-res 4637 |
This theorem is referenced by: (None) |
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