Theorem fsupprnfi 32588

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 8991	. 2 ⊢ { 0 } ∈ Fin
2		simpll 766	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → Fun 𝐹)
3		simplr 768	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → 𝐹 ∈ 𝑉)
4		simprl 770	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → 0 ∈ 𝑊)
5		ressupprn 32586	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
6	2, 3, 4, 5	syl3anc 1373	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
7		simprr 772	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → 𝐹 finSupp 0 )
8	7	fsuppimpd 9296	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → (𝐹 supp 0 ) ∈ Fin)
9		suppssdm 8133	. . . . . 6 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
10		ssdmres 5973	. . . . . 6 ⊢ ((𝐹 supp 0 ) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 ))
11	9, 10	mpbi 230	. . . . 5 ⊢ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 )
12	2	funresd 6543	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → Fun (𝐹 ↾ (𝐹 supp 0 )))
13		funforn 6761	. . . . . 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 218	. . . . 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 6751	. . . . . 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 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 )))
17	11, 14, 16	sylancr 587	. . . 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 9238	. . . 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 584	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
20	6, 19	eqeltrrd 2829	. 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → (ran 𝐹 ∖ { 0 }) ∈ Fin)
21		diffib 32423	. . 3 ⊢ ({ 0 } ∈ Fin → (ran 𝐹 ∈ Fin ↔ (ran 𝐹 ∖ { 0 }) ∈ Fin))
22	21	biimpar 477	. 2 ⊢ (({ 0 } ∈ Fin ∧ (ran 𝐹 ∖ { 0 }) ∈ Fin) → ran 𝐹 ∈ Fin)
23	1, 20, 22	sylancr 587	1 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran 𝐹 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3908   ⊆ wss 3911  {csn 4585   class class class wbr 5102  dom cdm 5631  ran crn 5632   ↾ cres 5633  Fun wfun 6493  –onto→wfo 6497  (class class class)co 7369   supp csupp 8116  Fincfn 8895   finSupp cfsupp 9288

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-supp 8117  df-1o 8411  df-en 8896  df-dom 8897  df-fin 8899  df-fsupp 9289

This theorem is referenced by:  elrspunidl  33372