![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5686 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
2 | cnvfi 9176 | . . 3 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin) | |
3 | dmfi 9326 | . . 3 ⊢ (◡𝐴 ∈ Fin → dom ◡𝐴 ∈ Fin) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ Fin → dom ◡𝐴 ∈ Fin) |
5 | 1, 4 | eqeltrid 2838 | 1 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ◡ccnv 5674 dom cdm 5675 ran crn 5676 Fincfn 8935 |
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 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
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 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7851 df-1st 7970 df-2nd 7971 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-fin 8939 |
This theorem is referenced by: f1dmvrnfibi 9332 unirnffid 9340 abrexfi 9348 gsum2dlem1 19830 gsum2dlem2 19831 tsmsxplem1 23639 prdsmet 23858 itg1addlem4 25198 relfi 31811 imafi2 31914 cmpcref 32768 carsggect 33255 carsgclctunlem2 33256 carsgclctunlem3 33257 breprexplema 33580 ptrecube 36426 heicant 36461 mblfinlem1 36463 ftc1anclem3 36501 istotbnd3 36577 sstotbnd2 36580 sstotbnd 36581 totbndbnd 36595 cantnfub 42004 cantnfub2 42005 rnmptfi 43800 rnffi 43804 choicefi 43832 stoweidlem39 44690 stoweidlem59 44710 fourierdlem31 44789 fourierdlem42 44800 fourierdlem54 44811 aacllem 47750 |
Copyright terms: Public domain | W3C validator |