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Mirrors > Home > MPE Home > Th. List > Mathboxes > rnffi | Structured version Visualization version GIF version |
Description: The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
rnffi | ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin) → ran 𝐹 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffi 42288 | . 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin) | |
2 | rnfi 8893 | . 2 ⊢ (𝐹 ∈ Fin → ran 𝐹 ∈ Fin) | |
3 | 1, 2 | syl 17 | 1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin) → ran 𝐹 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2114 ran crn 5536 ⟶wf 6346 Fincfn 8568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-om 7613 df-1st 7727 df-2nd 7728 df-1o 8144 df-er 8333 df-en 8569 df-dom 8570 df-fin 8572 |
This theorem is referenced by: fourierdlem70 43300 fourierdlem71 43301 fourierdlem80 43310 fourierdlem113 43343 hoicvr 43669 |
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