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Mirrors > Home > MPE Home > Th. List > fnresdm | Structured version Visualization version GIF version |
Description: A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.) |
Ref | Expression |
---|---|
fnresdm | ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 6651 | . 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | fndm 6652 | . . 3 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
3 | eqimss 4040 | . . 3 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 ⊆ 𝐴) |
5 | relssres 6022 | . 2 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ 𝐴) → (𝐹 ↾ 𝐴) = 𝐹) | |
6 | 1, 4, 5 | syl2anc 583 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ⊆ wss 3948 dom cdm 5676 ↾ cres 5678 Rel wrel 5681 Fn wfn 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-dm 5686 df-res 5688 df-fun 6545 df-fn 6546 |
This theorem is referenced by: fnima 6680 fresin 6760 resasplit 6761 fresaunres2 6763 fvreseq1 7040 fnsnr 7165 fninfp 7174 fnsnsplit 7184 fsnunfv 7187 fsnunres 7188 fnsuppeq0 8182 mapunen 9152 dif1enlem 9162 dif1enlemOLD 9163 fnfi 9187 canthp1lem2 10654 fseq1p1m1 13582 facnn 14242 fac0 14243 hashgval 14300 hashinf 14302 rlimres 15509 lo1res 15510 rlimresb 15516 isercolllem2 15619 isercoll 15621 ruclem4 16184 fsets 17109 sscres 17777 sscid 17778 gsumzres 19825 rnrhmsubrg 20503 pwssplit1 20902 zzngim 21417 ptuncnv 23630 ptcmpfi 23636 tsmsres 23967 imasdsf1olem 24198 tmslem 24309 tmslemOLD 24310 tmsxms 24314 imasf1oxms 24317 prdsxms 24358 tmsxps 24364 tmsxpsmopn 24365 isngp2 24425 tngngp2 24488 cnfldms 24611 cncms 25202 cnfldcusp 25204 mbfres2 25493 dvres 25759 dvres3a 25762 cpnres 25786 dvmptres3 25807 dvlip2 25847 dvgt0lem2 25855 dvne0 25863 rlimcnp2 26811 jensen 26833 eupthvdres 29920 sspg 30413 ssps 30415 sspn 30421 hhsssh 30954 fnresin 32282 padct 32376 ffsrn 32386 resf1o 32387 gsumle 32677 symgcom 32679 cycpmconjvlem 32735 cycpmconjslem1 32748 nsgqusf1o 32966 ply1degltdimlem 33160 cnrrext 33453 indf1ofs 33487 eulerpartlemt 33833 subfacp1lem3 34636 subfacp1lem5 34638 cvmliftlem11 34749 poimirlem9 36960 mapfzcons1 41917 eq0rabdioph 41976 eldioph4b 42011 diophren 42013 pwssplit4 42293 tfsconcatrev 42560 dvresntr 45092 sge0split 45583 |
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