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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 5560 | . 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) | |
2 | df-res 5560 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
3 | 2 | ineq1i 4182 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) |
4 | xpindir 5698 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V)) | |
5 | 4 | ineq2i 4183 | . . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) |
6 | df-res 5560 | . . 3 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) | |
7 | inass 4193 | . . 3 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) | |
8 | 5, 6, 7 | 3eqtr4ri 2852 | . 2 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
9 | 1, 3, 8 | 3eqtri 2845 | 1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 Vcvv 3492 ∩ cin 3932 × cxp 5546 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-xp 5554 df-rel 5555 df-res 5560 |
This theorem is referenced by: rescom 5872 resabs1 5876 resima2 5881 resmpt3 5899 resdisj 6019 rescnvcnv 6054 fresin 6540 resdif 6628 curry1 7788 curry2 7791 wfrlem4 7947 pmresg 8423 gruima 10212 rlimres 14903 lo1res 14904 rlimresb 14910 lo1eq 14913 rlimeq 14914 fsets 16504 setsid 16526 sscres 17081 gsumzres 18958 txkgen 22188 tsmsres 22679 ressxms 23062 ressms 23063 dvres 24436 dvres3a 24439 cpnres 24461 dvmptres3 24480 rlimcnp2 25471 df1stres 30365 df2ndres 30366 indf1ofs 31184 frrlem4 33023 dfrcl2 39897 relexpaddss 39941 limsupresuz 41860 liminfresuz 41941 fouriersw 42393 fouriercn 42394 |
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