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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 5689 | . 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) | |
2 | df-res 5689 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
3 | 2 | ineq1i 4209 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) |
4 | xpindir 5835 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V)) | |
5 | 4 | ineq2i 4210 | . . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) |
6 | df-res 5689 | . . 3 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) | |
7 | inass 4220 | . . 3 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) | |
8 | 5, 6, 7 | 3eqtr4ri 2772 | . 2 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
9 | 1, 3, 8 | 3eqtri 2765 | 1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3475 ∩ cin 3948 × cxp 5675 ↾ cres 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-opab 5212 df-xp 5683 df-rel 5684 df-res 5689 |
This theorem is referenced by: rescom 6008 resabs1 6012 resima2 6017 resmpt3 6039 resdisj 6169 rescnvcnv 6204 fresin 6761 resdif 6855 curry1 8090 curry2 8093 frrlem4 8274 wfrlem4OLD 8312 pmresg 8864 gruima 10797 rlimres 15502 lo1res 15503 rlimresb 15509 lo1eq 15512 rlimeq 15513 fsets 17102 setsid 17141 sscres 17770 gsumzres 19777 txkgen 23156 tsmsres 23648 ressxms 24034 ressms 24035 dvres 25428 dvres3a 25431 cpnres 25454 dvmptres3 25473 rlimcnp2 26471 df1stres 31925 df2ndres 31926 indf1ofs 33024 dfrcl2 42425 relexpaddss 42469 limsupresuz 44419 liminfresuz 44500 fouriersw 44947 fouriercn 44948 |
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