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Mirrors > Home > MPE Home > Th. List > rnfi | Structured version Visualization version GIF version |
Description: The range of a finite set is finite. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
rnfi | ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5560 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
2 | cnvfi 8795 | . . 3 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin) | |
3 | dmfi 8791 | . . 3 ⊢ (◡𝐴 ∈ Fin → dom ◡𝐴 ∈ Fin) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ Fin → dom ◡𝐴 ∈ Fin) |
5 | 1, 4 | eqeltrid 2917 | 1 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ◡ccnv 5548 dom cdm 5549 ran crn 5550 Fincfn 8498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7569 df-1st 7680 df-2nd 7681 df-1o 8093 df-er 8279 df-en 8499 df-dom 8500 df-fin 8502 |
This theorem is referenced by: f1dmvrnfibi 8797 unirnffid 8805 abrexfi 8813 gsum2dlem1 19021 gsum2dlem2 19022 tsmsxplem1 22690 prdsmet 22909 relfi 30281 imafi2 30374 cmpcref 31014 carsggect 31476 carsgclctunlem2 31477 carsgclctunlem3 31478 breprexplema 31801 ptrecube 34774 heicant 34809 mblfinlem1 34811 ftc1anclem3 34851 istotbnd3 34932 sstotbnd2 34935 sstotbnd 34936 totbndbnd 34950 rnmptfi 41307 rnffi 41311 choicefi 41343 stoweidlem39 42205 stoweidlem59 42225 fourierdlem31 42304 fourierdlem42 42315 fourierdlem54 42326 aacllem 44800 |
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