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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 5696 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
| 2 | cnvfi 9216 | . . 3 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin) | |
| 3 | dmfi 9375 | . . 3 ⊢ (◡𝐴 ∈ Fin → dom ◡𝐴 ∈ Fin) | |
| 4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ Fin → dom ◡𝐴 ∈ Fin) |
| 5 | 1, 4 | eqeltrid 2845 | 1 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ◡ccnv 5684 dom cdm 5685 ran crn 5686 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-pow 5365 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-mpt 5226 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-1st 8014 df-2nd 8015 df-1o 8506 df-en 8986 df-dom 8987 df-fin 8989 |
| This theorem is referenced by: f1dmvrnfibi 9381 unirnffid 9387 abrexfi 9392 gsum2dlem1 19988 gsum2dlem2 19989 tsmsxplem1 24161 prdsmet 24380 itg1addlem4 25734 relfi 32615 imafi2 32723 elrgspnsubrunlem1 33251 elrgspnsubrunlem2 33252 cmpcref 33849 carsggect 34320 carsgclctunlem2 34321 carsgclctunlem3 34322 breprexplema 34645 ptrecube 37627 heicant 37662 mblfinlem1 37664 ftc1anclem3 37702 istotbnd3 37778 sstotbnd2 37781 sstotbnd 37782 totbndbnd 37796 cantnfub 43334 cantnfub2 43335 rnmptfi 45176 rnffi 45180 choicefi 45205 stoweidlem39 46054 stoweidlem59 46074 fourierdlem31 46153 fourierdlem42 46164 fourierdlem54 46175 aacllem 49320 |
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