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Mirrors > Home > MPE Home > Th. List > imafi | Structured version Visualization version GIF version |
Description: Images of finite sets are finite. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
imafi | ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadmres 5544 | . 2 ⊢ (𝐹 “ dom (𝐹 ↾ 𝑋)) = (𝐹 “ 𝑋) | |
2 | simpr 476 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin) | |
3 | dmres 5339 | . . . . 5 ⊢ dom (𝐹 ↾ 𝑋) = (𝑋 ∩ dom 𝐹) | |
4 | inss1 3795 | . . . . 5 ⊢ (𝑋 ∩ dom 𝐹) ⊆ 𝑋 | |
5 | 3, 4 | eqsstri 3598 | . . . 4 ⊢ dom (𝐹 ↾ 𝑋) ⊆ 𝑋 |
6 | ssfi 8065 | . . . 4 ⊢ ((𝑋 ∈ Fin ∧ dom (𝐹 ↾ 𝑋) ⊆ 𝑋) → dom (𝐹 ↾ 𝑋) ∈ Fin) | |
7 | 2, 5, 6 | sylancl 693 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → dom (𝐹 ↾ 𝑋) ∈ Fin) |
8 | resss 5342 | . . . . 5 ⊢ (𝐹 ↾ 𝑋) ⊆ 𝐹 | |
9 | dmss 5245 | . . . . 5 ⊢ ((𝐹 ↾ 𝑋) ⊆ 𝐹 → dom (𝐹 ↾ 𝑋) ⊆ dom 𝐹) | |
10 | 8, 9 | mp1i 13 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → dom (𝐹 ↾ 𝑋) ⊆ dom 𝐹) |
11 | fores 6037 | . . . 4 ⊢ ((Fun 𝐹 ∧ dom (𝐹 ↾ 𝑋) ⊆ dom 𝐹) → (𝐹 ↾ dom (𝐹 ↾ 𝑋)):dom (𝐹 ↾ 𝑋)–onto→(𝐹 “ dom (𝐹 ↾ 𝑋))) | |
12 | 10, 11 | syldan 486 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 ↾ dom (𝐹 ↾ 𝑋)):dom (𝐹 ↾ 𝑋)–onto→(𝐹 “ dom (𝐹 ↾ 𝑋))) |
13 | fofi 8135 | . . 3 ⊢ ((dom (𝐹 ↾ 𝑋) ∈ Fin ∧ (𝐹 ↾ dom (𝐹 ↾ 𝑋)):dom (𝐹 ↾ 𝑋)–onto→(𝐹 “ dom (𝐹 ↾ 𝑋))) → (𝐹 “ dom (𝐹 ↾ 𝑋)) ∈ Fin) | |
14 | 7, 12, 13 | syl2anc 691 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ dom (𝐹 ↾ 𝑋)) ∈ Fin) |
15 | 1, 14 | syl5eqelr 2693 | 1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 dom cdm 5038 ↾ cres 5040 “ cima 5041 Fun wfun 5798 –onto→wfo 5802 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-fin 7845 |
This theorem is referenced by: fissuni 8154 fipreima 8155 fsuppcolem 8189 cmpfi 21021 mdegldg 23630 mdegcl 23633 locfinreflem 29235 sibfof 29729 eulerpartlemgf 29768 poimirlem30 32609 ftc1anclem7 32661 ftc1anc 32663 elrfirn 36276 sge0f1o 39275 trlsegvdeglem6 41393 |
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