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| Mirrors > Home > MPE Home > Th. List > fnssres | Structured version Visualization version GIF version | ||
| Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.) |
| Ref | Expression |
|---|---|
| fnssres | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fnssresb 6690 | . 2 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴)) | |
| 2 | 1 | biimpar 477 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ⊆ wss 3951 ↾ cres 5687 Fn wfn 6556 |
| 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-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: fnssresd 6692 fnresin1 6693 fnresin2 6694 fnresi 6697 fssres 6774 fvreseq0 7058 fnreseql 7068 ffvresb 7145 fnressn 7178 soisores 7347 oprres 7601 ofres 7716 fsplitfpar 8143 fnsuppres 8216 tfrlem1 8416 tz7.48lem 8481 tz7.49c 8486 resixp 8973 ixpfi2 9390 ttrclss 9760 dfac12lem1 10184 ackbij2lem3 10280 cfsmolem 10310 alephsing 10316 ttukeylem3 10551 iunfo 10579 fpwwe2lem7 10677 mulnzcnf 11909 seqfeq2 14066 seqf1olem2 14083 bpolylem 16084 reeff1 16156 sscres 17867 fullsubc 17895 fullresc 17896 funcres2c 17948 dmaf 18094 cdaf 18095 frmdplusg 18867 frmdss2 18876 gass 19319 dprdfadd 20040 rngmgpf 20154 mgpf 20245 prdscrngd 20319 rnghmresfn 20619 rnghmsscmap2 20629 rnghmsscmap 20630 rhmresfn 20648 rhmsscmap2 20658 rhmsscmap 20659 subrgascl 22090 upxp 23631 uptx 23633 cnmpt1st 23676 cnmpt2nd 23677 cnextfres1 24076 prdstmdd 24132 ressprdsds 24381 prdsxmslem2 24542 xrsdsre 24832 recosf1o 26577 resinf1o 26578 mpodvdsmulf1o 27237 dvdsmulf1o 27239 ex-fpar 30481 sspg 30747 ssps 30749 sspmlem 30751 sspn 30755 hhssnv 31283 ressupprn 32699 1stpreimas 32715 cnre2csqlem 33909 raddcn 33928 carsggect 34320 subiwrdlen 34388 signsvtn0 34585 signstres 34590 bnj1253 35031 bnj1280 35034 gblacfnacd 35113 subfacp1lem5 35189 cvmlift2lem9a 35308 filnetlem4 36382 finixpnum 37612 poimirlem4 37631 poimirlem8 37635 ftc1anclem3 37702 isdrngo2 37965 diaintclN 41060 dibintclN 41169 dihintcl 41346 imaiinfv 42704 fnwe2lem2 43063 aomclem6 43071 deg1mhm 43212 limsupvaluz2 45753 supcnvlimsup 45755 limsupgtlem 45792 resincncf 45890 icccncfext 45902 fourierdlem42 46164 fourierdlem73 46194 fdivmpt 48461 |
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