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Mirrors > Home > MPE Home > Th. List > rescom | Structured version Visualization version GIF version |
Description: Commutative law for restriction. (Contributed by NM, 27-Mar-1998.) |
Ref | Expression |
---|---|
rescom | ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4101 | . . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵) | |
2 | 1 | reseq2i 5833 | . 2 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ↾ (𝐶 ∩ 𝐵)) |
3 | resres 5849 | . 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶)) | |
4 | resres 5849 | . 2 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶 ∩ 𝐵)) | |
5 | 2, 3, 4 | 3eqtr4i 2769 | 1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∩ cin 3852 ↾ cres 5538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-opab 5102 df-xp 5542 df-rel 5543 df-res 5548 |
This theorem is referenced by: resabs2 5868 setscom 16709 dvres3a 24765 cpnres 24788 dvmptres3 24807 limsupresuz 42862 liminfresuz 42943 |
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