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Mirrors > Home > MPE Home > Th. List > dmfi | Structured version Visualization version GIF version |
Description: The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.) |
Ref | Expression |
---|---|
dmfi | ⊢ (𝐴 ∈ Fin → dom 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fidomdm 9324 | . 2 ⊢ (𝐴 ∈ Fin → dom 𝐴 ≼ 𝐴) | |
2 | domfi 9187 | . 2 ⊢ ((𝐴 ∈ Fin ∧ dom 𝐴 ≼ 𝐴) → dom 𝐴 ∈ Fin) | |
3 | 1, 2 | mpdan 686 | 1 ⊢ (𝐴 ∈ Fin → dom 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5146 dom cdm 5674 ≼ cdom 8932 Fincfn 8934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-om 7850 df-1st 7969 df-2nd 7970 df-1o 8460 df-er 8698 df-en 8935 df-dom 8936 df-fin 8938 |
This theorem is referenced by: fundmfibi 9326 residfi 9328 rnfi 9330 hashfun 14392 hashreshashfun 14394 psgnprfval 19381 gsum2dlem2 19830 gsum2d 19831 tsmsxp 23640 numedglnl 28383 vtxdginducedm1fi 28780 finsumvtxdg2ssteplem2 28782 finsumvtxdg2ssteplem4 28784 finsumvtxdg2sstep 28785 vtxdgoddnumeven 28789 relfi 31810 fedgmullem2 32659 esum2d 33028 etransclem27 44911 |
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