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Mirrors > Home > MPE Home > Th. List > resindir | Structured version Visualization version GIF version |
Description: Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.) |
Ref | Expression |
---|---|
resindir | ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inindir 4244 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V))) | |
2 | df-res 5701 | . 2 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) | |
3 | df-res 5701 | . . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
4 | df-res 5701 | . . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
5 | 3, 4 | ineq12i 4226 | . 2 ⊢ ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V))) |
6 | 1, 2, 5 | 3eqtr4i 2773 | 1 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3478 ∩ cin 3962 × cxp 5687 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-in 3970 df-res 5701 |
This theorem is referenced by: inimass 6177 fnreseql 7068 xrnres3 38386 |
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