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Mirrors > Home > ILE Home > Th. List > resindir | 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 3368 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V))) | |
2 | df-res 4653 | . 2 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ∩ 𝐵) ∩ (𝐶 × V)) | |
3 | df-res 4653 | . . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
4 | df-res 4653 | . . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
5 | 3, 4 | ineq12i 3349 | . 2 ⊢ ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V))) |
6 | 1, 2, 5 | 3eqtr4i 2220 | 1 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 Vcvv 2752 ∩ cin 3143 × cxp 4639 ↾ cres 4643 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-in 3150 df-res 4653 |
This theorem is referenced by: inimass 5060 fnreseql 5642 |
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