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Mirrors > Home > MPE Home > Th. List > fundmfibi | Structured version Visualization version GIF version |
Description: A function is finite if and only if its domain is finite. (Contributed by AV, 10-Jan-2020.) |
Ref | Expression |
---|---|
fundmfibi | ⊢ (Fun 𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmfi 9326 | . 2 ⊢ (𝐹 ∈ Fin → dom 𝐹 ∈ Fin) | |
2 | funfn 6575 | . . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
3 | fnfi 9177 | . . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ Fin) → 𝐹 ∈ Fin) | |
4 | 2, 3 | sylanb 582 | . . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ Fin) → 𝐹 ∈ Fin) |
5 | 4 | ex 414 | . 2 ⊢ (Fun 𝐹 → (dom 𝐹 ∈ Fin → 𝐹 ∈ Fin)) |
6 | 1, 5 | impbid2 225 | 1 ⊢ (Fun 𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2107 dom cdm 5675 Fun wfun 6534 Fn wfn 6535 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 negfi 12159 hashfundm 14398 vtxdgfusgrf 28734 tfsnfin 42035 fmtnoinf 46139 |
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