![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8781 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴) | |
2 | domfi 8723 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin) | |
3 | 1, 2 | syldan 594 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 –onto→wfo 6322 ≼ cdom 8490 Fincfn 8492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-fin 8496 |
This theorem is referenced by: f1fi 8795 imafi 8801 f1opwfi 8812 indexfi 8816 intrnfi 8864 infpwfien 9473 ttukeylem6 9925 fseqsupcl 13340 fiinfnf1o 13706 vdwlem6 16312 0ram2 16347 0ramcl 16349 mplsubrglem 20677 tgcmp 22006 hauscmplem 22011 1stcfb 22050 comppfsc 22137 1stckgenlem 22158 ptcnplem 22226 txtube 22245 txcmplem1 22246 tmdgsum2 22701 tsmsf1o 22750 tsmsxplem1 22758 ovolicc2lem4 24124 i1fadd 24299 i1fmul 24300 itg1addlem4 24303 i1fmulc 24307 mbfi1fseqlem4 24322 limciun 24497 edgusgrnbfin 27163 fsupprnfi 30452 erdszelem2 32552 mvrsfpw 32866 itg2addnclem2 35109 istotbnd3 35209 sstotbnd 35213 prdsbnd 35231 cntotbnd 35234 heiborlem1 35249 heibor 35259 lmhmfgima 40028 |
Copyright terms: Public domain | W3C validator |