![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mapfi | Structured version Visualization version GIF version |
Description: Set exponentiation of finite sets is finite. (Contributed by Jeff Madsen, 19-Jun-2011.) |
Ref | Expression |
---|---|
mapfi | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ↑m 𝐵) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpfi 9317 | . . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝐵 × 𝐴) ∈ Fin) | |
2 | 1 | ancoms 460 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐵 × 𝐴) ∈ Fin) |
3 | pwfi 9178 | . . 3 ⊢ ((𝐵 × 𝐴) ∈ Fin ↔ 𝒫 (𝐵 × 𝐴) ∈ Fin) | |
4 | 2, 3 | sylib 217 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → 𝒫 (𝐵 × 𝐴) ∈ Fin) |
5 | mapsspw 8872 | . 2 ⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) | |
6 | ssfi 9173 | . 2 ⊢ ((𝒫 (𝐵 × 𝐴) ∈ Fin ∧ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)) → (𝐴 ↑m 𝐵) ∈ Fin) | |
7 | 4, 5, 6 | sylancl 587 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ↑m 𝐵) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3949 𝒫 cpw 4603 × cxp 5675 (class class class)co 7409 ↑m cmap 8820 Fincfn 8939 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-1o 8466 df-map 8822 df-pm 8823 df-en 8940 df-fin 8943 |
This theorem is referenced by: ixpfi 9349 hashmap 14395 hashpw 14396 hashf1lem2 14417 prmreclem2 16850 vdwlem10 16923 efmndbasfi 18758 symgbasfi 19246 aannenlem1 25841 birthdaylem1 26456 dchrfi 26758 reprfi 33628 deranglem 34157 poimirlem9 36497 poimirlem26 36514 poimirlem27 36515 poimirlem28 36516 poimirlem32 36520 dvnprodlem2 44663 etransclem16 44966 etransclem33 44983 |
Copyright terms: Public domain | W3C validator |