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Mirrors > Home > MPE Home > Th. List > resfnfinfin | Structured version Visualization version GIF version |
Description: The restriction of a function to a finite set is finite. (Contributed by Alexander van der Vekens, 3-Feb-2018.) |
Ref | Expression |
---|---|
resfnfinfin | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ 𝐵) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 6650 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → Rel 𝐹) |
3 | resindm 6028 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹 ↾ 𝐵)) | |
4 | 3 | eqcomd 2733 | . . 3 ⊢ (Rel 𝐹 → (𝐹 ↾ 𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹))) |
5 | 2, 4 | syl 17 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ 𝐵) = (𝐹 ↾ (𝐵 ∩ dom 𝐹))) |
6 | fnfun 6648 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
7 | 6 | funfnd 6578 | . . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn dom 𝐹) |
8 | fnresin2 6675 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹)) | |
9 | infi 9282 | . . . . . 6 ⊢ (𝐵 ∈ Fin → (𝐵 ∩ dom 𝐹) ∈ Fin) | |
10 | fnfi 9195 | . . . . . 6 ⊢ (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ (𝐵 ∩ dom 𝐹) ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin) | |
11 | 9, 10 | sylan2 592 | . . . . 5 ⊢ (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ 𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin) |
12 | 11 | ex 412 | . . . 4 ⊢ ((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) → (𝐵 ∈ Fin → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)) |
13 | 7, 8, 12 | 3syl 18 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ Fin → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin)) |
14 | 13 | imp 406 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin) |
15 | 5, 14 | eqeltrd 2828 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ 𝐵) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3943 dom cdm 5672 ↾ cres 5674 Rel wrel 5677 Fn wfn 6537 Fincfn 8953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7732 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7863 df-1o 8478 df-en 8954 df-fin 8957 |
This theorem is referenced by: residfi 9347 itg1addlem4 25602 gsumhashmul 32735 pthhashvtx 34660 imadomfi 41397 |
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