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Mirrors > Home > MPE Home > Th. List > Mathboxes > imafi2 | Structured version Visualization version GIF version |
Description: The image by a finite set is finite. See also imafi 8806. (Contributed by Thierry Arnoux, 25-Apr-2020.) |
Ref | Expression |
---|---|
imafi2 | ⊢ (𝐴 ∈ Fin → (𝐴 “ 𝐵) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5562 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
2 | resss 5872 | . . . 4 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 | |
3 | ssfi 8727 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ↾ 𝐵) ⊆ 𝐴) → (𝐴 ↾ 𝐵) ∈ Fin) | |
4 | 2, 3 | mpan2 687 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ↾ 𝐵) ∈ Fin) |
5 | rnfi 8796 | . . 3 ⊢ ((𝐴 ↾ 𝐵) ∈ Fin → ran (𝐴 ↾ 𝐵) ∈ Fin) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 ∈ Fin → ran (𝐴 ↾ 𝐵) ∈ Fin) |
7 | 1, 6 | eqeltrid 2917 | 1 ⊢ (𝐴 ∈ Fin → (𝐴 “ 𝐵) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3935 ran crn 5550 ↾ cres 5551 “ cima 5552 Fincfn 8498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7569 df-1st 7680 df-2nd 7681 df-1o 8093 df-er 8279 df-en 8499 df-dom 8500 df-fin 8502 |
This theorem is referenced by: gsummpt2d 30615 esum2d 31252 |
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