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Mirrors > Home > ILE Home > Th. List > fundmfi | GIF version |
Description: The domain of a finite function is finite. (Contributed by Jim Kingdon, 5-Feb-2022.) |
Ref | Expression |
---|---|
fundmfi | ⊢ ((𝐴 ∈ Fin ∧ Fun 𝐴) → dom 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fundmeng 6454 | . 2 ⊢ ((𝐴 ∈ Fin ∧ Fun 𝐴) → dom 𝐴 ≈ 𝐴) | |
2 | enfii 6520 | . 2 ⊢ ((𝐴 ∈ Fin ∧ dom 𝐴 ≈ 𝐴) → dom 𝐴 ∈ Fin) | |
3 | 1, 2 | syldan 276 | 1 ⊢ ((𝐴 ∈ Fin ∧ Fun 𝐴) → dom 𝐴 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1434 class class class wbr 3811 dom cdm 4401 Fun wfun 4963 ≈ cen 6385 Fincfn 6387 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-sbc 2827 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-er 6222 df-en 6388 df-fin 6390 |
This theorem is referenced by: fundmfibi 6573 funrnfi 6576 |
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