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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 8785 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴) | |
2 | domfi 8727 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin) | |
3 | 1, 2 | syldan 591 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 class class class wbr 5057 –onto→wfo 6346 ≼ cdom 8495 Fincfn 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-fin 8501 |
This theorem is referenced by: f1fi 8799 imafi 8805 f1opwfi 8816 indexfi 8820 intrnfi 8868 infpwfien 9476 ttukeylem6 9924 fseqsupcl 13333 fiinfnf1o 13698 vdwlem6 16310 0ram2 16345 0ramcl 16347 mplsubrglem 20147 tgcmp 21937 hauscmplem 21942 1stcfb 21981 comppfsc 22068 1stckgenlem 22089 ptcnplem 22157 txtube 22176 txcmplem1 22177 tmdgsum2 22632 tsmsf1o 22680 tsmsxplem1 22688 ovolicc2lem4 24048 i1fadd 24223 i1fmul 24224 itg1addlem4 24227 i1fmulc 24231 mbfi1fseqlem4 24246 limciun 24419 edgusgrnbfin 27082 erdszelem2 32336 mvrsfpw 32650 itg2addnclem2 34825 istotbnd3 34930 sstotbnd 34934 prdsbnd 34952 cntotbnd 34955 heiborlem1 34970 heibor 34980 lmhmfgima 39562 |
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