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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 9350 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴) | |
| 2 | domfi 9229 | . 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 395 ∈ wcel 2108 class class class wbr 5143 –onto→wfo 6559 ≼ cdom 8983 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-en 8986 df-dom 8987 df-fin 8989 |
| This theorem is referenced by: f1fi 9352 imafi 9353 f1opwfi 9396 indexfi 9400 intrnfi 9456 infpwfien 10102 ttukeylem6 10554 fseqsupcl 14018 fiinfnf1o 14389 vdwlem6 17024 0ram2 17059 0ramcl 17061 mplsubrglem 22024 tgcmp 23409 hauscmplem 23414 1stcfb 23453 comppfsc 23540 1stckgenlem 23561 ptcnplem 23629 txtube 23648 txcmplem1 23649 tmdgsum2 24104 tsmsf1o 24153 tsmsxplem1 24161 ovolicc2lem4 25555 i1fadd 25730 i1fmul 25731 itg1addlem4 25734 i1fmulc 25738 mbfi1fseqlem4 25753 limciun 25929 edgusgrnbfin 29390 fsupprnfi 32701 erdszelem2 35197 mvrsfpw 35511 itg2addnclem2 37679 istotbnd3 37778 sstotbnd 37782 prdsbnd 37800 cntotbnd 37803 heiborlem1 37818 heibor 37828 lmhmfgima 43096 |
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