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Theorem ffsrn 29632
 Description: The range of a finitely supported function is finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
Hypotheses
Ref Expression
ffsrn.z (𝜑𝑍𝑊)
ffsrn.0 (𝜑𝐹𝑉)
ffsrn.1 (𝜑 → Fun 𝐹)
ffsrn.2 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
Assertion
Ref Expression
ffsrn (𝜑 → ran 𝐹 ∈ Fin)

Proof of Theorem ffsrn
StepHypRef Expression
1 imaundi 5580 . . . . . . 7 (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍}))
21reseq2i 5425 . . . . . 6 (𝐹 ↾ (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍})))
3 undif1 4076 . . . . . . . . 9 ((V ∖ {𝑍}) ∪ {𝑍}) = (V ∪ {𝑍})
4 ssv 3658 . . . . . . . . . 10 {𝑍} ⊆ V
5 ssequn2 3819 . . . . . . . . . 10 ({𝑍} ⊆ V ↔ (V ∪ {𝑍}) = V)
64, 5mpbi 220 . . . . . . . . 9 (V ∪ {𝑍}) = V
73, 6eqtri 2673 . . . . . . . 8 ((V ∖ {𝑍}) ∪ {𝑍}) = V
87imaeq2i 5499 . . . . . . 7 (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = (𝐹 “ V)
98reseq2i 5425 . . . . . 6 (𝐹 ↾ (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ (𝐹 “ V))
10 resundi 5445 . . . . . 6 (𝐹 ↾ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍}))) = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍})))
112, 9, 103eqtr3i 2681 . . . . 5 (𝐹 ↾ (𝐹 “ V)) = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍})))
12 ffsrn.1 . . . . . 6 (𝜑 → Fun 𝐹)
13 dfdm4 5348 . . . . . . 7 dom 𝐹 = ran 𝐹
14 dfrn4 5630 . . . . . . 7 ran 𝐹 = (𝐹 “ V)
1513, 14eqtri 2673 . . . . . 6 dom 𝐹 = (𝐹 “ V)
16 df-fn 5929 . . . . . . 7 (𝐹 Fn (𝐹 “ V) ↔ (Fun 𝐹 ∧ dom 𝐹 = (𝐹 “ V)))
17 fnresdm 6038 . . . . . . 7 (𝐹 Fn (𝐹 “ V) → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
1816, 17sylbir 225 . . . . . 6 ((Fun 𝐹 ∧ dom 𝐹 = (𝐹 “ V)) → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
1912, 15, 18sylancl 695 . . . . 5 (𝜑 → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
2011, 19syl5reqr 2700 . . . 4 (𝜑𝐹 = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))))
2120rneqd 5385 . . 3 (𝜑 → ran 𝐹 = ran ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))))
22 rnun 5576 . . 3 ran ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))) = (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍})))
2321, 22syl6eq 2701 . 2 (𝜑 → ran 𝐹 = (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍}))))
24 ffsrn.0 . . . . . 6 (𝜑𝐹𝑉)
25 ffsrn.z . . . . . 6 (𝜑𝑍𝑊)
26 suppimacnv 7351 . . . . . 6 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
2724, 25, 26syl2anc 694 . . . . 5 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
28 ffsrn.2 . . . . 5 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
2927, 28eqeltrrd 2731 . . . 4 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ Fin)
30 cnvexg 7154 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
31 imaexg 7145 . . . . . 6 (𝐹 ∈ V → (𝐹 “ (V ∖ {𝑍})) ∈ V)
3224, 30, 313syl 18 . . . . 5 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ V)
33 cnvimass 5520 . . . . . . 7 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
34 fores 6162 . . . . . . 7 ((Fun 𝐹 ∧ (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹) → (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))):(𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (𝐹 “ (V ∖ {𝑍}))))
3512, 33, 34sylancl 695 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))):(𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (𝐹 “ (V ∖ {𝑍}))))
36 fofn 6155 . . . . . 6 ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))):(𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (𝐹 “ (V ∖ {𝑍}))) → (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) Fn (𝐹 “ (V ∖ {𝑍})))
3735, 36syl 17 . . . . 5 (𝜑 → (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) Fn (𝐹 “ (V ∖ {𝑍})))
38 fnrndomg 9396 . . . . 5 ((𝐹 “ (V ∖ {𝑍})) ∈ V → ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) Fn (𝐹 “ (V ∖ {𝑍})) → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ≼ (𝐹 “ (V ∖ {𝑍}))))
3932, 37, 38sylc 65 . . . 4 (𝜑 → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ≼ (𝐹 “ (V ∖ {𝑍})))
40 domfi 8222 . . . 4 (((𝐹 “ (V ∖ {𝑍})) ∈ Fin ∧ ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ≼ (𝐹 “ (V ∖ {𝑍}))) → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
4129, 39, 40syl2anc 694 . . 3 (𝜑 → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
42 snfi 8079 . . . 4 {𝑍} ∈ Fin
43 df-ima 5156 . . . . . 6 (𝐹 “ (𝐹 “ {𝑍})) = ran (𝐹 ↾ (𝐹 “ {𝑍}))
44 funimacnv 6008 . . . . . . 7 (Fun 𝐹 → (𝐹 “ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
4512, 44syl 17 . . . . . 6 (𝜑 → (𝐹 “ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
4643, 45syl5eqr 2699 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
47 inss1 3866 . . . . 5 ({𝑍} ∩ ran 𝐹) ⊆ {𝑍}
4846, 47syl6eqss 3688 . . . 4 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) ⊆ {𝑍})
49 ssfi 8221 . . . 4 (({𝑍} ∈ Fin ∧ ran (𝐹 ↾ (𝐹 “ {𝑍})) ⊆ {𝑍}) → ran (𝐹 ↾ (𝐹 “ {𝑍})) ∈ Fin)
5042, 48, 49sylancr 696 . . 3 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) ∈ Fin)
51 unfi 8268 . . 3 ((ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin ∧ ran (𝐹 ↾ (𝐹 “ {𝑍})) ∈ Fin) → (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍}))) ∈ Fin)
5241, 50, 51syl2anc 694 . 2 (𝜑 → (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍}))) ∈ Fin)
5323, 52eqeltrd 2730 1 (𝜑 → ran 𝐹 ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604   ∪ cun 3605   ∩ cin 3606   ⊆ wss 3607  {csn 4210   class class class wbr 4685  ◡ccnv 5142  dom cdm 5143  ran crn 5144   ↾ cres 5145   “ cima 5146  Fun wfun 5920   Fn wfn 5921  –onto→wfo 5924  (class class class)co 6690   supp csupp 7340   ≼ cdom 7995  Fincfn 7997 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-ac2 9323 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-fin 8001  df-card 8803  df-acn 8806  df-ac 8977 This theorem is referenced by:  fpwrelmapffslem  29635
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