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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 8406 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴) | |
2 | domfi 8348 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin) | |
3 | 1, 2 | syldan 488 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 2139 class class class wbr 4804 –onto→wfo 6047 ≼ cdom 8121 Fincfn 8123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-om 7232 df-1o 7730 df-er 7913 df-en 8124 df-dom 8125 df-fin 8127 |
This theorem is referenced by: f1fi 8420 imafi 8426 f1opwfi 8437 indexfi 8441 intrnfi 8489 infpwfien 9095 ttukeylem6 9548 fseqsupcl 12990 fiinfnf1o 13352 vdwlem6 15912 0ram2 15947 0ramcl 15949 mplsubrglem 19661 tgcmp 21426 hauscmplem 21431 1stcfb 21470 comppfsc 21557 1stckgenlem 21578 ptcnplem 21646 txtube 21665 txcmplem1 21666 tmdgsum2 22121 tsmsf1o 22169 tsmsxplem1 22177 ovolicc2lem4 23508 i1fadd 23681 i1fmul 23682 itg1addlem4 23685 i1fmulc 23689 mbfi1fseqlem4 23704 limciun 23877 edgusgrnbfin 26494 erdszelem2 31502 mvrsfpw 31731 itg2addnclem2 33793 istotbnd3 33901 sstotbnd 33905 prdsbnd 33923 cntotbnd 33926 heiborlem1 33941 heibor 33951 lmhmfgima 38174 |
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