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Mirrors > Home > MPE Home > Th. List > fofi | Structured version Visualization version GIF version |
Description: If an onto function has a finite domain, its codomain/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 9351 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴) | |
2 | domfi 9217 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin) | |
3 | 1, 2 | syldan 589 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 class class class wbr 5149 –onto→wfo 6547 ≼ cdom 8962 Fincfn 8964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-om 7872 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-fin 8968 |
This theorem is referenced by: f1fi 9365 imafiALT 9371 f1opwfi 9382 indexfi 9386 intrnfi 9441 infpwfien 10087 ttukeylem6 10539 fseqsupcl 13978 fiinfnf1o 14345 vdwlem6 16958 0ram2 16993 0ramcl 16995 mplsubrglem 21966 tgcmp 23349 hauscmplem 23354 1stcfb 23393 comppfsc 23480 1stckgenlem 23501 ptcnplem 23569 txtube 23588 txcmplem1 23589 tmdgsum2 24044 tsmsf1o 24093 tsmsxplem1 24101 ovolicc2lem4 25493 i1fadd 25668 i1fmul 25669 itg1addlem4 25672 itg1addlem4OLD 25673 i1fmulc 25677 mbfi1fseqlem4 25692 limciun 25867 edgusgrnbfin 29258 fsupprnfi 32554 erdszelem2 34933 mvrsfpw 35247 itg2addnclem2 37276 istotbnd3 37375 sstotbnd 37379 prdsbnd 37397 cntotbnd 37400 heiborlem1 37415 heibor 37425 lmhmfgima 42650 |
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