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Theorem fsupprnfi 32701
Description: Finite support implies finite range. (Contributed by Thierry Arnoux, 24-Jun-2024.)
Assertion
Ref Expression
fsupprnfi (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → ran 𝐹 ∈ Fin)

Proof of Theorem fsupprnfi
StepHypRef Expression
1 snfi 9083 . 2 { 0 } ∈ Fin
2 simpll 767 . . . 4 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → Fun 𝐹)
3 simplr 769 . . . 4 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → 𝐹𝑉)
4 simprl 771 . . . 4 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → 0𝑊)
5 ressupprn 32699 . . . 4 ((Fun 𝐹𝐹𝑉0𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
62, 3, 4, 5syl3anc 1373 . . 3 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
7 simprr 773 . . . . 5 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → 𝐹 finSupp 0 )
87fsuppimpd 9409 . . . 4 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → (𝐹 supp 0 ) ∈ Fin)
9 suppssdm 8202 . . . . . 6 (𝐹 supp 0 ) ⊆ dom 𝐹
10 ssdmres 6031 . . . . . 6 ((𝐹 supp 0 ) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
119, 10mpbi 230 . . . . 5 dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 )
122funresd 6609 . . . . . 6 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → Fun (𝐹 ↾ (𝐹 supp 0 )))
13 funforn 6827 . . . . . 6 (Fun (𝐹 ↾ (𝐹 supp 0 )) ↔ (𝐹 ↾ (𝐹 supp 0 )):dom (𝐹 ↾ (𝐹 supp 0 ))–onto→ran (𝐹 ↾ (𝐹 supp 0 )))
1412, 13sylib 218 . . . . 5 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → (𝐹 ↾ (𝐹 supp 0 )):dom (𝐹 ↾ (𝐹 supp 0 ))–onto→ran (𝐹 ↾ (𝐹 supp 0 )))
15 foeq2 6817 . . . . . 6 (dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ) → ((𝐹 ↾ (𝐹 supp 0 )):dom (𝐹 ↾ (𝐹 supp 0 ))–onto→ran (𝐹 ↾ (𝐹 supp 0 )) ↔ (𝐹 ↾ (𝐹 supp 0 )):(𝐹 supp 0 )–onto→ran (𝐹 ↾ (𝐹 supp 0 ))))
1615biimpa 476 . . . . 5 ((dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ) ∧ (𝐹 ↾ (𝐹 supp 0 )):dom (𝐹 ↾ (𝐹 supp 0 ))–onto→ran (𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 ↾ (𝐹 supp 0 )):(𝐹 supp 0 )–onto→ran (𝐹 ↾ (𝐹 supp 0 )))
1711, 14, 16sylancr 587 . . . 4 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → (𝐹 ↾ (𝐹 supp 0 )):(𝐹 supp 0 )–onto→ran (𝐹 ↾ (𝐹 supp 0 )))
18 fofi 9351 . . . 4 (((𝐹 supp 0 ) ∈ Fin ∧ (𝐹 ↾ (𝐹 supp 0 )):(𝐹 supp 0 )–onto→ran (𝐹 ↾ (𝐹 supp 0 ))) → ran (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
198, 17, 18syl2anc 584 . . 3 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → ran (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
206, 19eqeltrrd 2842 . 2 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → (ran 𝐹 ∖ { 0 }) ∈ Fin)
21 diffib 32540 . . 3 ({ 0 } ∈ Fin → (ran 𝐹 ∈ Fin ↔ (ran 𝐹 ∖ { 0 }) ∈ Fin))
2221biimpar 477 . 2 (({ 0 } ∈ Fin ∧ (ran 𝐹 ∖ { 0 }) ∈ Fin) → ran 𝐹 ∈ Fin)
231, 20, 22sylancr 587 1 (((Fun 𝐹𝐹𝑉) ∧ ( 0𝑊𝐹 finSupp 0 )) → ran 𝐹 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cdif 3948  wss 3951  {csn 4626   class class class wbr 5143  dom cdm 5685  ran crn 5686  cres 5687  Fun wfun 6555  ontowfo 6559  (class class class)co 7431   supp csupp 8185  Fincfn 8985   finSupp cfsupp 9401
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-rep 5279  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-csb 3900  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-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  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-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-supp 8186  df-1o 8506  df-en 8986  df-dom 8987  df-fin 8989  df-fsupp 9402
This theorem is referenced by:  elrspunidl  33456
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