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Mirrors > Home > MPE Home > Th. List > fssres | Structured version Visualization version GIF version |
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.) |
Ref | Expression |
---|---|
fssres | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 6539 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | fnssres 6663 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶) Fn 𝐶) | |
3 | resss 6001 | . . . . . . 7 ⊢ (𝐹 ↾ 𝐶) ⊆ 𝐹 | |
4 | 3 | rnssi 5934 | . . . . . 6 ⊢ ran (𝐹 ↾ 𝐶) ⊆ ran 𝐹 |
5 | sstr 3988 | . . . . . 6 ⊢ ((ran (𝐹 ↾ 𝐶) ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → ran (𝐹 ↾ 𝐶) ⊆ 𝐵) | |
6 | 4, 5 | mpan 689 | . . . . 5 ⊢ (ran 𝐹 ⊆ 𝐵 → ran (𝐹 ↾ 𝐶) ⊆ 𝐵) |
7 | 2, 6 | anim12i 614 | . . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴) ∧ ran 𝐹 ⊆ 𝐵) → ((𝐹 ↾ 𝐶) Fn 𝐶 ∧ ran (𝐹 ↾ 𝐶) ⊆ 𝐵)) |
8 | 7 | an32s 651 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) ∧ 𝐶 ⊆ 𝐴) → ((𝐹 ↾ 𝐶) Fn 𝐶 ∧ ran (𝐹 ↾ 𝐶) ⊆ 𝐵)) |
9 | 1, 8 | sylanb 582 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → ((𝐹 ↾ 𝐶) Fn 𝐶 ∧ ran (𝐹 ↾ 𝐶) ⊆ 𝐵)) |
10 | df-f 6539 | . 2 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶𝐵 ↔ ((𝐹 ↾ 𝐶) Fn 𝐶 ∧ ran (𝐹 ↾ 𝐶) ⊆ 𝐵)) | |
11 | 9, 10 | sylibr 233 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ⊆ wss 3946 ran crn 5673 ↾ cres 5674 Fn wfn 6530 ⟶wf 6531 |
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-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
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-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-fun 6537 df-fn 6538 df-f 6539 |
This theorem is referenced by: fssresd 6748 fssres2 6749 fresin 6750 fresaun 6752 f1ssres 6785 f2ndf 8093 elmapssres 8849 pmresg 8852 ralxpmap 8878 mapunen 9134 fofinf1o 9315 fseqenlem1 10006 inar1 10757 gruima 10784 addnqf 10930 mulnqf 10931 fseq1p1m1 13562 injresinj 13740 seqf1olem2 13995 wrdred1 14497 rlimres 15489 lo1res 15490 vdwnnlem1 16915 fsets 17089 resmhm 18688 resghm 19093 gsumzres 19760 gsumzadd 19773 gsum2dlem2 19822 dpjidcl 19911 ablfac1eu 19926 abvres 20424 znf1o 21080 islindf4 21366 kgencn 23029 ptrescn 23112 hmeores 23244 tsmsres 23617 tsmsmhm 23619 tsmsadd 23620 xrge0gsumle 24318 xrge0tsms 24319 ovolicc2lem4 25006 limcdif 25362 limcflf 25367 limcmo 25368 dvres 25397 dvres3a 25400 aannenlem1 25810 logcn 26124 dvlog 26128 dvlog2 26130 logtayl 26137 dvatan 26407 atancn 26408 efrlim 26441 amgm 26462 dchrelbas2 26707 redwlklem 28895 pthdivtx 28953 hhssabloilem 30479 hhssnv 30482 wrdres 32074 gsumpart 32178 xrge0tsmsd 32180 cntmeas 33155 eulerpartlemt 33301 eulerpartlemmf 33305 eulerpartlemgvv 33306 subiwrd 33315 sseqp1 33325 poimirlem4 36397 mbfresfi 36439 mbfposadd 36440 itg2gt0cn 36448 sdclem2 36516 mzpcompact2lem 41360 eldiophb 41366 eldioph2 41371 cncfiooicclem1 44482 fouriersw 44820 sge0tsms 44969 psmeasure 45060 sssmf 45327 resmgmhm 46441 lindslinindimp2lem2 46980 |
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