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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 5049 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
2 | cnvfi 8131 | . . 3 ⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin) | |
3 | dmfi 8129 | . . 3 ⊢ (◡𝐴 ∈ Fin → dom ◡𝐴 ∈ Fin) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ Fin → dom ◡𝐴 ∈ Fin) |
5 | 1, 4 | syl5eqel 2692 | 1 ⊢ (𝐴 ∈ Fin → ran 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ◡ccnv 5037 dom cdm 5038 ran crn 5039 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1st 7059 df-2nd 7060 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-fin 7845 |
This theorem is referenced by: f1dmvrnfibi 8133 unirnffid 8141 abrexfi 8149 gsum2dlem1 18192 gsum2dlem2 18193 tsmsxplem1 21766 prdsmet 21985 usgrafiedg 25945 cusgrafi 26010 sizeusglecusg 26014 relfi 28797 imafi2 28872 cmpcref 29245 carsggect 29707 carsgclctunlem2 29708 carsgclctunlem3 29709 ptrecube 32579 heicant 32614 mblfinlem1 32616 ftc1anclem3 32657 istotbnd3 32740 sstotbnd2 32743 sstotbnd 32744 totbndbnd 32758 rnmptfi 38346 rnffi 38351 choicefi 38387 stoweidlem39 38932 stoweidlem59 38952 fourierdlem31 39031 fourierdlem42 39042 fourierdlem54 39053 aacllem 42356 |
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