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Mirrors > Home > MPE Home > Th. List > fndmfifsupp | Structured version Visualization version GIF version |
Description: A function with a finite domain is always finitely supported. (Contributed by AV, 25-May-2019.) |
Ref | Expression |
---|---|
fndmfisuppfi.f | β’ (π β πΉ Fn π·) |
fndmfisuppfi.d | β’ (π β π· β Fin) |
fndmfisuppfi.z | β’ (π β π β π) |
Ref | Expression |
---|---|
fndmfifsupp | β’ (π β πΉ finSupp π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fndmfisuppfi.f | . . 3 β’ (π β πΉ Fn π·) | |
2 | dffn3 6664 | . . 3 β’ (πΉ Fn π· β πΉ:π·βΆran πΉ) | |
3 | 1, 2 | sylib 217 | . 2 β’ (π β πΉ:π·βΆran πΉ) |
4 | fndmfisuppfi.d | . 2 β’ (π β π· β Fin) | |
5 | fndmfisuppfi.z | . 2 β’ (π β π β π) | |
6 | 3, 4, 5 | fdmfifsupp 9236 | 1 β’ (π β πΉ finSupp π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2105 class class class wbr 5092 ran crn 5621 Fn wfn 6474 βΆwf 6475 Fincfn 8804 finSupp cfsupp 9226 |
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-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-supp 8048 df-1o 8367 df-en 8805 df-fin 8808 df-fsupp 9227 |
This theorem is referenced by: resfifsupp 9254 gsummptcl 19663 esumgsum 32311 gsumesum 32325 matunitlindflem1 35886 matunitlindflem2 35887 |
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