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Mirrors > Home > MPE Home > Th. List > fofi | Structured version Visualization version GIF version |
Description: If a function has a finite domain, its range is finite. Theorem 37 of [Suppes] p. 104. (Contributed by NM, 25-Mar-2007.) |
Ref | Expression |
---|---|
fofi | ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fodomfi 8239 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴) | |
2 | domfi 8181 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin) | |
3 | 1, 2 | syldan 487 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 –onto→wfo 5886 ≼ cdom 7953 Fincfn 7955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-fin 7959 |
This theorem is referenced by: f1fi 8253 imafi 8259 f1opwfi 8270 indexfi 8274 intrnfi 8322 infpwfien 8885 ttukeylem6 9336 fseqsupcl 12776 fiinfnf1o 13138 vdwlem6 15690 0ram2 15725 0ramcl 15727 mplsubrglem 19439 tgcmp 21204 hauscmplem 21209 1stcfb 21248 comppfsc 21335 1stckgenlem 21356 ptcnplem 21424 txtube 21443 txcmplem1 21444 tmdgsum2 21900 tsmsf1o 21948 tsmsxplem1 21956 ovolicc2lem4 23288 i1fadd 23462 i1fmul 23463 itg1addlem4 23466 i1fmulc 23470 mbfi1fseqlem4 23485 limciun 23658 edgusgrnbfin 26275 erdszelem2 31174 mvrsfpw 31403 itg2addnclem2 33462 istotbnd3 33570 sstotbnd 33574 prdsbnd 33592 cntotbnd 33595 heiborlem1 33610 heibor 33620 lmhmfgima 37654 |
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