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| 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 9358 | . . . 4 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → (𝐵 × 𝐴) ∈ Fin) | |
| 2 | 1 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐵 × 𝐴) ∈ Fin) |
| 3 | pwfi 9357 | . . 3 ⊢ ((𝐵 × 𝐴) ∈ Fin ↔ 𝒫 (𝐵 × 𝐴) ∈ Fin) | |
| 4 | 2, 3 | sylib 218 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → 𝒫 (𝐵 × 𝐴) ∈ Fin) |
| 5 | mapsspw 8918 | . 2 ⊢ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) | |
| 6 | ssfi 9213 | . 2 ⊢ ((𝒫 (𝐵 × 𝐴) ∈ Fin ∧ (𝐴 ↑m 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)) → (𝐴 ↑m 𝐵) ∈ Fin) | |
| 7 | 4, 5, 6 | sylancl 586 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ↑m 𝐵) ∈ Fin) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 𝒫 cpw 4600 × cxp 5683 (class class class)co 7431 ↑m cmap 8866 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-1o 8506 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-fin 8989 |
| This theorem is referenced by: ixpfi 9389 hashmap 14474 hashpw 14475 hashf1lem2 14495 prmreclem2 16955 vdwlem10 17028 efmndbasfi 18890 symgbasfi 19396 aannenlem1 26370 birthdaylem1 26994 dchrfi 27299 reprfi 34631 deranglem 35171 poimirlem9 37636 poimirlem26 37653 poimirlem27 37654 poimirlem28 37655 poimirlem32 37659 dvnprodlem2 45962 etransclem16 46265 etransclem33 46282 |
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