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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 5367 | . 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) | |
2 | df-res 5367 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
3 | 2 | ineq1i 4033 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) |
4 | xpindir 5502 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V)) | |
5 | 4 | ineq2i 4034 | . . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) |
6 | df-res 5367 | . . 3 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) | |
7 | inass 4044 | . . 3 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) | |
8 | 5, 6, 7 | 3eqtr4ri 2813 | . 2 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
9 | 1, 3, 8 | 3eqtri 2806 | 1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 Vcvv 3398 ∩ 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-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 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-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-opab 4949 df-xp 5361 df-rel 5362 df-res 5367 |
This theorem is referenced by: rescom 5672 resabs1 5676 resima2 5681 resmpt3 5700 resdisj 5817 rescnvcnv 5851 fresin 6323 resdif 6411 curry1 7550 curry2 7553 wfrlem4 7700 wfrlem4OLD 7701 pmresg 8168 gruima 9959 rlimres 14697 lo1res 14698 rlimresb 14704 lo1eq 14707 rlimeq 14708 fsets 16288 setsid 16310 sscres 16868 gsumzres 18696 txkgen 21864 tsmsres 22355 ressxms 22738 ressms 22739 dvres 24112 dvres3a 24115 cpnres 24137 dvmptres3 24156 rlimcnp2 25145 df1stres 30047 df2ndres 30048 indf1ofs 30686 frrlem4 32372 dfrcl2 38927 relexpaddss 38971 limsupresuz 40847 liminfresuz 40928 fouriersw 41379 fouriercn 41380 |
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