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Mirrors > Home > MPE Home > Th. List > resdifcom | Structured version Visualization version GIF version |
Description: Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.) |
Ref | Expression |
---|---|
resdifcom | ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indif1 4270 | . 2 ⊢ ((𝐴 ∖ 𝐶) ∩ (𝐵 × V)) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶) | |
2 | df-res 5690 | . 2 ⊢ ((𝐴 ∖ 𝐶) ↾ 𝐵) = ((𝐴 ∖ 𝐶) ∩ (𝐵 × V)) | |
3 | df-res 5690 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
4 | 3 | difeq1i 4114 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶) |
5 | 1, 2, 4 | 3eqtr4ri 2764 | 1 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3461 ∖ cdif 3941 ∩ cin 3943 × cxp 5676 ↾ cres 5680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-in 3951 df-res 5690 |
This theorem is referenced by: setsfun0 17144 cycpmrn 32956 tocyccntz 32957 |
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