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Theorem ffsrn 29370
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 5509 . . . . . . 7 (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍}))
21reseq2i 5358 . . . . . 6 (𝐹 ↾ (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍})))
3 undif1 4020 . . . . . . . . 9 ((V ∖ {𝑍}) ∪ {𝑍}) = (V ∪ {𝑍})
4 ssv 3609 . . . . . . . . . 10 {𝑍} ⊆ V
5 ssequn2 3769 . . . . . . . . . 10 ({𝑍} ⊆ V ↔ (V ∪ {𝑍}) = V)
64, 5mpbi 220 . . . . . . . . 9 (V ∪ {𝑍}) = V
73, 6eqtri 2643 . . . . . . . 8 ((V ∖ {𝑍}) ∪ {𝑍}) = V
87imaeq2i 5428 . . . . . . 7 (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = (𝐹 “ V)
98reseq2i 5358 . . . . . 6 (𝐹 ↾ (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ (𝐹 “ V))
10 resundi 5374 . . . . . 6 (𝐹 ↾ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍}))) = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍})))
112, 9, 103eqtr3i 2651 . . . . 5 (𝐹 ↾ (𝐹 “ V)) = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍})))
12 ffsrn.1 . . . . . 6 (𝜑 → Fun 𝐹)
13 dfdm4 5281 . . . . . . 7 dom 𝐹 = ran 𝐹
14 dfrn4 5559 . . . . . . 7 ran 𝐹 = (𝐹 “ V)
1513, 14eqtri 2643 . . . . . 6 dom 𝐹 = (𝐹 “ V)
16 df-fn 5855 . . . . . . 7 (𝐹 Fn (𝐹 “ V) ↔ (Fun 𝐹 ∧ dom 𝐹 = (𝐹 “ V)))
17 fnresdm 5963 . . . . . . 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 693 . . . . 5 (𝜑 → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
2011, 19syl5reqr 2670 . . . 4 (𝜑𝐹 = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))))
2120rneqd 5318 . . 3 (𝜑 → ran 𝐹 = ran ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))))
22 rnun 5505 . . 3 ran ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))) = (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍})))
2321, 22syl6eq 2671 . 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 7258 . . . . . 6 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
2724, 25, 26syl2anc 692 . . . . 5 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
28 ffsrn.2 . . . . 5 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
2927, 28eqeltrrd 2699 . . . 4 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ Fin)
30 cnvexg 7066 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
31 imaexg 7057 . . . . . 6 (𝐹 ∈ V → (𝐹 “ (V ∖ {𝑍})) ∈ V)
3224, 30, 313syl 18 . . . . 5 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ V)
33 cnvimass 5449 . . . . . . 7 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
34 fores 6086 . . . . . . 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 6079 . . . . . 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 9310 . . . . 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 8133 . . . 4 (((𝐹 “ (V ∖ {𝑍})) ∈ Fin ∧ ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ≼ (𝐹 “ (V ∖ {𝑍}))) → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
4129, 39, 40syl2anc 692 . . 3 (𝜑 → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
42 snfi 7990 . . . 4 {𝑍} ∈ Fin
43 df-ima 5092 . . . . . 6 (𝐹 “ (𝐹 “ {𝑍})) = ran (𝐹 ↾ (𝐹 “ {𝑍}))
44 funimacnv 5933 . . . . . . 7 (Fun 𝐹 → (𝐹 “ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
4512, 44syl 17 . . . . . 6 (𝜑 → (𝐹 “ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
4643, 45syl5eqr 2669 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
47 inss1 3816 . . . . 5 ({𝑍} ∩ ran 𝐹) ⊆ {𝑍}
4846, 47syl6eqss 3639 . . . 4 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) ⊆ {𝑍})
49 ssfi 8132 . . . 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 8179 . . 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 692 . 2 (𝜑 → (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍}))) ∈ Fin)
5323, 52eqeltrd 2698 1 (𝜑 → ran 𝐹 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  {csn 4153   class class class wbr 4618  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  Fun wfun 5846   Fn wfn 5847  ontowfo 5850  (class class class)co 6610   supp csupp 7247  cdom 7905  Fincfn 7907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-ac2 9237
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-fin 7911  df-card 8717  df-acn 8720  df-ac 8891
This theorem is referenced by:  fpwrelmapffslem  29373
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