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Mirrors > Home > MPE Home > Th. List > resres | Structured version Visualization version GIF version |
Description: The restriction of a restriction. (Contributed by NM, 27-Mar-2008.) |
Ref | Expression |
---|---|
resres | ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5531 | . 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) | |
2 | df-res 5531 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
3 | 2 | ineq1i 4135 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) |
4 | xpindir 5669 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V)) | |
5 | 4 | ineq2i 4136 | . . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) |
6 | df-res 5531 | . . 3 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) | |
7 | inass 4146 | . . 3 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) | |
8 | 5, 6, 7 | 3eqtr4ri 2832 | . 2 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
9 | 1, 3, 8 | 3eqtri 2825 | 1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 Vcvv 3441 ∩ cin 3880 × cxp 5517 ↾ cres 5521 |
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-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 df-rel 5526 df-res 5531 |
This theorem is referenced by: rescom 5844 resabs1 5848 resima2 5853 resmpt3 5873 resdisj 5993 rescnvcnv 6028 fresin 6521 resdif 6610 curry1 7782 curry2 7785 wfrlem4 7941 pmresg 8417 gruima 10213 rlimres 14907 lo1res 14908 rlimresb 14914 lo1eq 14917 rlimeq 14918 fsets 16508 setsid 16530 sscres 17085 gsumzres 19022 txkgen 22257 tsmsres 22749 ressxms 23132 ressms 23133 dvres 24514 dvres3a 24517 cpnres 24540 dvmptres3 24559 rlimcnp2 25552 df1stres 30463 df2ndres 30464 indf1ofs 31395 frrlem4 33239 dfrcl2 40375 relexpaddss 40419 limsupresuz 42345 liminfresuz 42426 fouriersw 42873 fouriercn 42874 |
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