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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 8481 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴) | |
2 | domfi 8423 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin) | |
3 | 1, 2 | syldan 586 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 class class class wbr 4843 –onto→wfo 6099 ≼ cdom 8193 Fincfn 8195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-om 7300 df-1o 7799 df-er 7982 df-en 8196 df-dom 8197 df-fin 8199 |
This theorem is referenced by: f1fi 8495 imafi 8501 f1opwfi 8512 indexfi 8516 intrnfi 8564 infpwfien 9171 ttukeylem6 9624 fseqsupcl 13031 fiinfnf1o 13390 vdwlem6 16023 0ram2 16058 0ramcl 16060 mplsubrglem 19762 tgcmp 21533 hauscmplem 21538 1stcfb 21577 comppfsc 21664 1stckgenlem 21685 ptcnplem 21753 txtube 21772 txcmplem1 21773 tmdgsum2 22228 tsmsf1o 22276 tsmsxplem1 22284 ovolicc2lem4 23628 i1fadd 23803 i1fmul 23804 itg1addlem4 23807 i1fmulc 23811 mbfi1fseqlem4 23826 limciun 23999 edgusgrnbfin 26617 erdszelem2 31691 mvrsfpw 31920 itg2addnclem2 33950 istotbnd3 34057 sstotbnd 34061 prdsbnd 34079 cntotbnd 34082 heiborlem1 34097 heibor 34107 lmhmfgima 38439 |
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