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Theorem ffsrn 28892
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 5464 . . . . . . 7 (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍}))
21reseq2i 5314 . . . . . 6 (𝐹 ↾ (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍})))
3 undif1 3995 . . . . . . . . 9 ((V ∖ {𝑍}) ∪ {𝑍}) = (V ∪ {𝑍})
4 ssv 3588 . . . . . . . . . 10 {𝑍} ⊆ V
5 ssequn2 3748 . . . . . . . . . 10 ({𝑍} ⊆ V ↔ (V ∪ {𝑍}) = V)
64, 5mpbi 219 . . . . . . . . 9 (V ∪ {𝑍}) = V
73, 6eqtri 2632 . . . . . . . 8 ((V ∖ {𝑍}) ∪ {𝑍}) = V
87imaeq2i 5383 . . . . . . 7 (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = (𝐹 “ V)
98reseq2i 5314 . . . . . 6 (𝐹 ↾ (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ (𝐹 “ V))
10 resundi 5330 . . . . . 6 (𝐹 ↾ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍}))) = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍})))
112, 9, 103eqtr3i 2640 . . . . 5 (𝐹 ↾ (𝐹 “ V)) = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍})))
12 ffsrn.1 . . . . . 6 (𝜑 → Fun 𝐹)
13 dfdm4 5238 . . . . . . 7 dom 𝐹 = ran 𝐹
14 dfrn4 5513 . . . . . . 7 ran 𝐹 = (𝐹 “ V)
1513, 14eqtri 2632 . . . . . 6 dom 𝐹 = (𝐹 “ V)
16 df-fn 5807 . . . . . . 7 (𝐹 Fn (𝐹 “ V) ↔ (Fun 𝐹 ∧ dom 𝐹 = (𝐹 “ V)))
17 fnresdm 5914 . . . . . . 7 (𝐹 Fn (𝐹 “ V) → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
1816, 17sylbir 224 . . . . . 6 ((Fun 𝐹 ∧ dom 𝐹 = (𝐹 “ V)) → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
1912, 15, 18sylancl 693 . . . . 5 (𝜑 → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
2011, 19syl5reqr 2659 . . . 4 (𝜑𝐹 = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))))
2120rneqd 5274 . . 3 (𝜑 → ran 𝐹 = ran ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))))
22 rnun 5460 . . 3 ran ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))) = (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍})))
2321, 22syl6eq 2660 . 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 7193 . . . . . 6 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
2724, 25, 26syl2anc 691 . . . . 5 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
28 ffsrn.2 . . . . 5 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
2927, 28eqeltrrd 2689 . . . 4 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ Fin)
30 cnvexg 7005 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
31 imaexg 6995 . . . . . 6 (𝐹 ∈ V → (𝐹 “ (V ∖ {𝑍})) ∈ V)
3224, 30, 313syl 18 . . . . 5 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ V)
33 cnvimass 5404 . . . . . . 7 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
34 fores 6037 . . . . . . 7 ((Fun 𝐹 ∧ (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹) → (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))):(𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (𝐹 “ (V ∖ {𝑍}))))
3512, 33, 34sylancl 693 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))):(𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (𝐹 “ (V ∖ {𝑍}))))
36 fofn 6030 . . . . . 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 9239 . . . . 5 ((𝐹 “ (V ∖ {𝑍})) ∈ V → ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) Fn (𝐹 “ (V ∖ {𝑍})) → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ≼ (𝐹 “ (V ∖ {𝑍}))))
3932, 37, 38sylc 63 . . . 4 (𝜑 → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ≼ (𝐹 “ (V ∖ {𝑍})))
40 domfi 8066 . . . 4 (((𝐹 “ (V ∖ {𝑍})) ∈ Fin ∧ ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ≼ (𝐹 “ (V ∖ {𝑍}))) → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
4129, 39, 40syl2anc 691 . . 3 (𝜑 → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
42 snfi 7923 . . . 4 {𝑍} ∈ Fin
43 df-ima 5051 . . . . . 6 (𝐹 “ (𝐹 “ {𝑍})) = ran (𝐹 ↾ (𝐹 “ {𝑍}))
44 funimacnv 5884 . . . . . . 7 (Fun 𝐹 → (𝐹 “ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
4512, 44syl 17 . . . . . 6 (𝜑 → (𝐹 “ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
4643, 45syl5eqr 2658 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
47 inss1 3795 . . . . 5 ({𝑍} ∩ ran 𝐹) ⊆ {𝑍}
4846, 47syl6eqss 3618 . . . 4 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) ⊆ {𝑍})
49 ssfi 8065 . . . 4 (({𝑍} ∈ Fin ∧ ran (𝐹 ↾ (𝐹 “ {𝑍})) ⊆ {𝑍}) → ran (𝐹 ↾ (𝐹 “ {𝑍})) ∈ Fin)
5042, 48, 49sylancr 694 . . 3 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) ∈ Fin)
51 unfi 8112 . . 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 691 . 2 (𝜑 → (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍}))) ∈ Fin)
5323, 52eqeltrd 2688 1 (𝜑 → ran 𝐹 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cdif 3537  cun 3538  cin 3539  wss 3540  {csn 4125   class class class wbr 4583  ccnv 5037  dom cdm 5038  ran crn 5039  cres 5040  cima 5041  Fun wfun 5798   Fn wfn 5799  ontowfo 5802  (class class class)co 6549   supp csupp 7182  cdom 7839  Fincfn 7841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-ac2 9168
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-fin 7845  df-card 8648  df-acn 8651  df-ac 8822
This theorem is referenced by:  fpwrelmapffslem  28895
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