Theorem fsupprnfi 32783

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 8984	. 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 32781	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
6	2, 3, 4, 5	syl3anc 1374	. . 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 9276	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → (𝐹 supp 0 ) ∈ Fin)
9		suppssdm 8121	. . . . . 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 6536	. . . . . 6 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → Fun (𝐹 ↾ (𝐹 supp 0 )))
13		funforn 6754	. . . . . 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 6744	. . . . . 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 588	. . . 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 9217	. . . 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 585	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran (𝐹 ↾ (𝐹 supp 0 )) ∈ Fin)
20	6, 19	eqeltrrd 2838	. 2 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → (ran 𝐹 ∖ { 0 }) ∈ Fin)
21		diffib 32609	. . 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 588	1 ⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran 𝐹 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887   ⊆ wss 3890  {csn 4568   class class class wbr 5086  dom cdm 5625  ran crn 5626   ↾ cres 5627  Fun wfun 6487  –onto→wfo 6491  (class class class)co 7361   supp csupp 8104  Fincfn 8887   finSupp cfsupp 9268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-supp 8105  df-1o 8399  df-en 8888  df-dom 8889  df-fin 8891  df-fsupp 9269

This theorem is referenced by:  elrspunidl  33506