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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 4098 | . 2 ⊢ ((𝐴 ∖ 𝐶) ∩ (𝐵 × V)) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶) | |
2 | df-res 5367 | . 2 ⊢ ((𝐴 ∖ 𝐶) ↾ 𝐵) = ((𝐴 ∖ 𝐶) ∩ (𝐵 × V)) | |
3 | df-res 5367 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
4 | 3 | difeq1i 3947 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶) |
5 | 1, 2, 4 | 3eqtr4ri 2813 | 1 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 Vcvv 3398 ∖ cdif 3789 ∩ cin 3791 × cxp 5353 ↾ cres 5357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rab 3099 df-v 3400 df-dif 3795 df-in 3799 df-res 5367 |
This theorem is referenced by: setsfun0 16291 |
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