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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 8786 | . 2 ⊢ (𝐹 ∈ Fin → dom 𝐹 ∈ Fin) | |
2 | funfn 6354 | . . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
3 | fnfi 8780 | . . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ Fin) → 𝐹 ∈ Fin) | |
4 | 2, 3 | sylanb 584 | . . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ Fin) → 𝐹 ∈ Fin) |
5 | 4 | ex 416 | . 2 ⊢ (Fun 𝐹 → (dom 𝐹 ∈ Fin → 𝐹 ∈ Fin)) |
6 | 1, 5 | impbid2 229 | 1 ⊢ (Fun 𝐹 → (𝐹 ∈ Fin ↔ dom 𝐹 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 dom cdm 5519 Fun wfun 6318 Fn wfn 6319 Fincfn 8492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-fin 8496 |
This theorem is referenced by: f1dmvrnfibi 8792 negfi 11577 vtxdgfusgrf 27287 hashfundm 32464 fmtnoinf 44053 |
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