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Mirrors > Home > NFE Home > Th. List > resindir | GIF version |
Description: Class restriction distributes over intersection. (Contributed by set.mm contributors, 18-Dec-2008.) |
Ref | Expression |
---|---|
resindir | ⊢ ((A ∩ B) ↾ C) = ((A ↾ C) ∩ (B ↾ C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inindir 3473 | . 2 ⊢ ((A ∩ B) ∩ (C × V)) = ((A ∩ (C × V)) ∩ (B ∩ (C × V))) | |
2 | df-res 4788 | . 2 ⊢ ((A ∩ B) ↾ C) = ((A ∩ B) ∩ (C × V)) | |
3 | df-res 4788 | . . 3 ⊢ (A ↾ C) = (A ∩ (C × V)) | |
4 | df-res 4788 | . . 3 ⊢ (B ↾ C) = (B ∩ (C × V)) | |
5 | 3, 4 | ineq12i 3455 | . 2 ⊢ ((A ↾ C) ∩ (B ↾ C)) = ((A ∩ (C × V)) ∩ (B ∩ (C × V))) |
6 | 1, 2, 5 | 3eqtr4i 2383 | 1 ⊢ ((A ∩ B) ↾ C) = ((A ↾ C) ∩ (B ↾ C)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 Vcvv 2859 ∩ cin 3208 × cxp 4770 ↾ cres 4774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-res 4788 |
This theorem is referenced by: (None) |
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