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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 6254 | . 2 ⊢ (𝐹 “ dom (𝐹 ↾ 𝑋)) = (𝐹 “ 𝑋) | |
| 2 | simpr 484 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → 𝑋 ∈ Fin) | |
| 3 | dmres 6030 | . . . . 5 ⊢ dom (𝐹 ↾ 𝑋) = (𝑋 ∩ dom 𝐹) | |
| 4 | inss1 4237 | . . . . 5 ⊢ (𝑋 ∩ dom 𝐹) ⊆ 𝑋 | |
| 5 | 3, 4 | eqsstri 4030 | . . . 4 ⊢ dom (𝐹 ↾ 𝑋) ⊆ 𝑋 |
| 6 | ssfi 9213 | . . . 4 ⊢ ((𝑋 ∈ Fin ∧ dom (𝐹 ↾ 𝑋) ⊆ 𝑋) → dom (𝐹 ↾ 𝑋) ∈ Fin) | |
| 7 | 2, 5, 6 | sylancl 586 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → dom (𝐹 ↾ 𝑋) ∈ Fin) |
| 8 | resss 6019 | . . . . 5 ⊢ (𝐹 ↾ 𝑋) ⊆ 𝐹 | |
| 9 | dmss 5913 | . . . . 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 6830 | . . . 4 ⊢ ((Fun 𝐹 ∧ dom (𝐹 ↾ 𝑋) ⊆ dom 𝐹) → (𝐹 ↾ dom (𝐹 ↾ 𝑋)):dom (𝐹 ↾ 𝑋)–onto→(𝐹 “ dom (𝐹 ↾ 𝑋))) | |
| 12 | 10, 11 | syldan 591 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 ↾ dom (𝐹 ↾ 𝑋)):dom (𝐹 ↾ 𝑋)–onto→(𝐹 “ dom (𝐹 ↾ 𝑋))) |
| 13 | fofi 9351 | . . 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 584 | . 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ dom (𝐹 ↾ 𝑋)) ∈ Fin) |
| 15 | 1, 14 | eqeltrrid 2846 | 1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 dom cdm 5685 ↾ cres 5687 “ cima 5688 Fun wfun 6555 –onto→wfo 6559 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-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-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-br 5144 df-opab 5206 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-om 7888 df-1o 8506 df-en 8986 df-dom 8987 df-fin 8989 |
| This theorem is referenced by: pwfir 9355 pwfilem 9356 fissuni 9397 fipreima 9398 fsuppcolem 9441 cmpfi 23416 mdegldg 26105 mdegcl 26108 madefi 27950 oldfi 27951 trlsegvdeglem6 30244 fsuppcurry1 32736 fsuppcurry2 32737 elrgspnlem2 33247 elrgspnsubrunlem2 33252 elrspunidl 33456 locfinreflem 33839 zarcmplem 33880 sibfof 34342 eulerpartlemgf 34381 fineqvrep 35109 poimirlem30 37657 ftc1anclem7 37706 ftc1anc 37708 aks6d1c2 42131 aks6d1c6lem5 42178 elrfirn 42706 sge0f1o 46397 |
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