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Mirrors > Home > MPE Home > Th. List > resdifdi | Structured version Visualization version GIF version |
Description: Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.) |
Ref | Expression |
---|---|
resdifdi | ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5644 | . . 3 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = (𝐴 ∩ ((𝐵 ∖ 𝐶) × V)) | |
2 | difxp1 6116 | . . . 4 ⊢ ((𝐵 ∖ 𝐶) × V) = ((𝐵 × V) ∖ (𝐶 × V)) | |
3 | 2 | ineq2i 4168 | . . 3 ⊢ (𝐴 ∩ ((𝐵 ∖ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V))) |
4 | indifdi 4242 | . . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V))) | |
5 | 1, 3, 4 | 3eqtri 2768 | . 2 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V))) |
6 | df-res 5644 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
7 | df-res 5644 | . . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
8 | 6, 7 | difeq12i 4079 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V))) |
9 | 5, 8 | eqtr4i 2767 | 1 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 Vcvv 3444 ∖ cdif 3906 ∩ cin 3908 × cxp 5630 ↾ cres 5634 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-xp 5638 df-rel 5639 df-cnv 5640 df-res 5644 |
This theorem is referenced by: (None) |
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