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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 5697 | . 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) | |
| 2 | df-res 5697 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
| 3 | 2 | ineq1i 4216 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) |
| 4 | xpindir 5845 | . . . 4 ⊢ ((𝐵 ∩ 𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V)) | |
| 5 | 4 | ineq2i 4217 | . . 3 ⊢ (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) |
| 6 | df-res 5697 | . . 3 ⊢ (𝐴 ↾ (𝐵 ∩ 𝐶)) = (𝐴 ∩ ((𝐵 ∩ 𝐶) × V)) | |
| 7 | inass 4228 | . . 3 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) | |
| 8 | 5, 6, 7 | 3eqtr4ri 2776 | . 2 ⊢ ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
| 9 | 1, 3, 8 | 3eqtri 2769 | 1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵 ∩ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 Vcvv 3480 ∩ cin 3950 × cxp 5683 ↾ cres 5687 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 df-rel 5692 df-res 5697 |
| This theorem is referenced by: rescom 6020 resabs1 6024 resima2 6034 resmpt3 6056 resdisj 6189 rescnvcnv 6224 fresin 6777 resdif 6869 curry1 8129 curry2 8132 frrlem4 8314 wfrlem4OLD 8352 pmresg 8910 gruima 10842 rlimres 15594 lo1res 15595 rlimresb 15601 lo1eq 15604 rlimeq 15605 fsets 17206 setsid 17244 sscres 17867 gsumzres 19927 txkgen 23660 tsmsres 24152 ressxms 24538 ressms 24539 dvres 25946 dvres3a 25949 cpnres 25973 dvmptres3 25994 rlimcnp2 27009 df1stres 32713 df2ndres 32714 indf1ofs 32851 dfrcl2 43687 relexpaddss 43731 limsupresuz 45718 liminfresuz 45799 fouriersw 46246 fouriercn 46247 tposresg 48778 tposres3 48781 |
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