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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 6648 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → Rel 𝐹) |
3 | resindm 6028 | . . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) = (𝐹 ↾ 𝐵)) | |
4 | 3 | eqcomd 2738 | . . 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 6646 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
7 | 6 | funfnd 6576 | . . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹 Fn dom 𝐹) |
8 | fnresin2 6673 | . . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹)) | |
9 | infi 9264 | . . . . . 6 ⊢ (𝐵 ∈ Fin → (𝐵 ∩ dom 𝐹) ∈ Fin) | |
10 | fnfi 9177 | . . . . . 6 ⊢ (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ (𝐵 ∩ dom 𝐹) ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin) | |
11 | 9, 10 | sylan2 593 | . . . . 5 ⊢ (((𝐹 ↾ (𝐵 ∩ dom 𝐹)) Fn (𝐵 ∩ dom 𝐹) ∧ 𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin) |
12 | 11 | ex 413 | . . . 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 407 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ (𝐵 ∩ dom 𝐹)) ∈ Fin) |
15 | 5, 14 | eqeltrd 2833 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ Fin) → (𝐹 ↾ 𝐵) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3946 dom cdm 5675 ↾ cres 5677 Rel wrel 5680 Fn wfn 6535 Fincfn 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7852 df-1o 8462 df-en 8936 df-fin 8939 |
This theorem is referenced by: residfi 9329 itg1addlem4 25207 gsumhashmul 32195 pthhashvtx 34106 |
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