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Mirrors > Home > MPE Home > Th. List > resdifdir | Structured version Visualization version GIF version |
Description: Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.) |
Ref | Expression |
---|---|
resdifdir | ⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indifdir 4191 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V))) | |
2 | df-res 5540 | . 2 ⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ∖ 𝐵) ∩ (𝐶 × V)) | |
3 | df-res 5540 | . . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
4 | df-res 5540 | . . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
5 | 3, 4 | difeq12i 4028 | . 2 ⊢ ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V))) |
6 | 1, 2, 5 | 3eqtr4i 2791 | 1 ⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 Vcvv 3409 ∖ cdif 3857 ∩ cin 3859 × cxp 5526 ↾ cres 5530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2729 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-rab 3079 df-v 3411 df-dif 3863 df-in 3867 df-res 5540 |
This theorem is referenced by: (None) |
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