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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 5561 | . 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) | |
2 | df-res 5561 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
3 | 2 | ineq1i 4184 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) |
4 | xpindir 5699 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V)) | |
5 | 4 | ineq2i 4185 | . . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) |
6 | df-res 5561 | . . 3 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) | |
7 | inass 4195 | . . 3 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) | |
8 | 5, 6, 7 | 3eqtr4ri 2855 | . 2 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
9 | 1, 3, 8 | 3eqtri 2848 | 1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3494 ∩ cin 3934 × cxp 5547 ↾ cres 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5121 df-xp 5555 df-rel 5556 df-res 5561 |
This theorem is referenced by: rescom 5873 resabs1 5877 resima2 5882 resmpt3 5900 resdisj 6020 rescnvcnv 6055 fresin 6541 resdif 6629 curry1 7793 curry2 7796 wfrlem4 7952 pmresg 8428 gruima 10218 rlimres 14909 lo1res 14910 rlimresb 14916 lo1eq 14919 rlimeq 14920 fsets 16510 setsid 16532 sscres 17087 gsumzres 19023 txkgen 22254 tsmsres 22746 ressxms 23129 ressms 23130 dvres 24503 dvres3a 24506 cpnres 24528 dvmptres3 24547 rlimcnp2 25538 df1stres 30433 df2ndres 30434 indf1ofs 31280 frrlem4 33121 dfrcl2 40012 relexpaddss 40056 limsupresuz 41977 liminfresuz 42058 fouriersw 42510 fouriercn 42511 |
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