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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 9333 | . 2 ⊢ (𝐹 ∈ Fin → dom 𝐹 ∈ Fin) | |
2 | funfn 6578 | . . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
3 | fnfi 9184 | . . . 4 ⊢ ((𝐹 Fn dom 𝐹 ∧ dom 𝐹 ∈ Fin) → 𝐹 ∈ Fin) | |
4 | 2, 3 | sylanb 580 | . . 3 ⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ Fin) → 𝐹 ∈ Fin) |
5 | 4 | ex 412 | . 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 2105 dom cdm 5676 Fun wfun 6537 Fn wfn 6538 Fincfn 8942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7859 df-1st 7978 df-2nd 7979 df-1o 8469 df-er 8706 df-en 8943 df-dom 8944 df-fin 8946 |
This theorem is referenced by: f1dmvrnfibi 9339 negfi 12168 hashfundm 14407 vtxdgfusgrf 29022 tfsnfin 42405 fmtnoinf 46503 |
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