Theorem fsupprnfi 30535

Description: Finite support implies finite range. (Contributed by Thierry Arnoux, 24-Jun-2024.)

Assertion

Ref	Expression
fsupprnfi	⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran 𝐹 ∈ Fin)

Proof of Theorem fsupprnfi

Step	Hyp	Ref	Expression
1		snfi 8607	. 2 ⊢ { 0 } ∈ Fin
2		simpll 767	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → Fun 𝐹)
3		simplr 769	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → 𝐹 ∈ 𝑉)
4		simprl 771	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → 0 ∈ 𝑊)
5		ressupprn 30533	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
6	2, 3, 4, 5	syl3anc 1369	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
7		simprr 773	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → 𝐹 finSupp 0 )
8	7	fsuppimpd 8858	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → (𝐹 supp 0 ) ∈ Fin)
9		suppssdm 7844	. . . . . 6 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
10		ssdmres 5839	. . . . . 6 ⊢ ((𝐹 supp 0 ) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
11	9, 10	mpbi 233	. . . . 5 ⊢ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 )
12	2	funresd 6371	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → Fun (𝐹 ↾ (𝐹 supp 0 )))
13		funforn 6576	. . . . . 6 ⊢ (Fun (𝐹 ↾ (𝐹 supp 0 )) ↔ (𝐹 ↾ (𝐹 supp 0 )):dom (𝐹 ↾ (𝐹 supp 0 ))–onto→ran (𝐹 ↾ (𝐹 supp 0 )))
14	12, 13	sylib 221	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → (𝐹 ↾ (𝐹 supp 0 )):dom (𝐹 ↾ (𝐹 supp 0 ))–onto→ran (𝐹 ↾ (𝐹 supp 0 )))
15		foeq2 6566	. . . . . 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 ))))
16	15	biimpa 481	. . . . 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 )))
17	11, 14, 16	sylancr 591	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → (𝐹 ↾ (𝐹 supp 0 )):(𝐹 supp 0 )–onto→ran (𝐹 ↾ (𝐹 supp 0 )))
18		fofi 8828	. . . 4 ⊢ (((𝐹 supp 0 ) ∈ Fin ∧ (𝐹 ↾ (𝐹 supp 0 )):(𝐹 supp 0 )–onto→ran (𝐹 ↾ (𝐹 supp 0 ))) → ran (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
19	8, 17, 18	syl2anc 588	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
20	6, 19	eqeltrrd 2852	. 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → (ran 𝐹 ∖ { 0 }) ∈ Fin)
21		diffib 30376	. . 3 ⊢ ({ 0 } ∈ Fin → (ran 𝐹 ∈ Fin ↔ (ran 𝐹 ∖ { 0 }) ∈ Fin))
22	21	biimpar 482	. 2 ⊢ (({ 0 } ∈ Fin ∧ (ran 𝐹 ∖ { 0 }) ∈ Fin) → ran 𝐹 ∈ Fin)
23	1, 20, 22	sylancr 591	1 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran 𝐹 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ∖ cdif 3851   ⊆ wss 3854  {csn 4515   class class class wbr 5025  dom cdm 5517  ran crn 5518   ↾ cres 5519  Fun wfun 6322  –onto→wfo 6326  (class class class)co 7143   supp csupp 7828  Fincfn 8520   finSupp cfsupp 8851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5149  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-reu 3075  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-int 4832  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7146  df-oprab 7147  df-mpo 7148  df-om 7573  df-supp 7829  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-en 8521  df-dom 8522  df-fin 8524  df-fsupp 8852

This theorem is referenced by:  elrspunidl  31112