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Theorem ffsrn 30153
 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 5884 . . . . . . 7 (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍}))
21reseq2i 5731 . . . . . 6 (𝐹 ↾ (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍})))
3 undif1 4338 . . . . . . . . 9 ((V ∖ {𝑍}) ∪ {𝑍}) = (V ∪ {𝑍})
4 ssv 3912 . . . . . . . . . 10 {𝑍} ⊆ V
5 ssequn2 4080 . . . . . . . . . 10 ({𝑍} ⊆ V ↔ (V ∪ {𝑍}) = V)
64, 5mpbi 231 . . . . . . . . 9 (V ∪ {𝑍}) = V
73, 6eqtri 2819 . . . . . . . 8 ((V ∖ {𝑍}) ∪ {𝑍}) = V
87imaeq2i 5804 . . . . . . 7 (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = (𝐹 “ V)
98reseq2i 5731 . . . . . 6 (𝐹 ↾ (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ (𝐹 “ V))
10 resundi 5748 . . . . . 6 (𝐹 ↾ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍}))) = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍})))
112, 9, 103eqtr3i 2827 . . . . 5 (𝐹 ↾ (𝐹 “ V)) = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍})))
12 ffsrn.1 . . . . . 6 (𝜑 → Fun 𝐹)
13 dfdm4 5650 . . . . . . 7 dom 𝐹 = ran 𝐹
14 dfrn4 5934 . . . . . . 7 ran 𝐹 = (𝐹 “ V)
1513, 14eqtri 2819 . . . . . 6 dom 𝐹 = (𝐹 “ V)
16 df-fn 6228 . . . . . . 7 (𝐹 Fn (𝐹 “ V) ↔ (Fun 𝐹 ∧ dom 𝐹 = (𝐹 “ V)))
17 fnresdm 6336 . . . . . . 7 (𝐹 Fn (𝐹 “ V) → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
1816, 17sylbir 236 . . . . . 6 ((Fun 𝐹 ∧ dom 𝐹 = (𝐹 “ V)) → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
1912, 15, 18sylancl 586 . . . . 5 (𝜑 → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
2011, 19syl5reqr 2846 . . . 4 (𝜑𝐹 = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))))
2120rneqd 5690 . . 3 (𝜑 → ran 𝐹 = ran ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))))
22 rnun 5880 . . 3 ran ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))) = (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍})))
2321, 22syl6eq 2847 . 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 7692 . . . . . 6 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
2724, 25, 26syl2anc 584 . . . . 5 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
28 ffsrn.2 . . . . 5 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
2927, 28eqeltrrd 2884 . . . 4 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ Fin)
30 cnvexg 7485 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
31 imaexg 7476 . . . . . 6 (𝐹 ∈ V → (𝐹 “ (V ∖ {𝑍})) ∈ V)
3224, 30, 313syl 18 . . . . 5 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ V)
33 cnvimass 5825 . . . . . . 7 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
34 fores 6468 . . . . . . 7 ((Fun 𝐹 ∧ (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹) → (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))):(𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (𝐹 “ (V ∖ {𝑍}))))
3512, 33, 34sylancl 586 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))):(𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (𝐹 “ (V ∖ {𝑍}))))
36 fofn 6460 . . . . . 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 9804 . . . . 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 8585 . . . 4 (((𝐹 “ (V ∖ {𝑍})) ∈ Fin ∧ ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ≼ (𝐹 “ (V ∖ {𝑍}))) → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
4129, 39, 40syl2anc 584 . . 3 (𝜑 → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
42 snfi 8442 . . . 4 {𝑍} ∈ Fin
43 df-ima 5456 . . . . . 6 (𝐹 “ (𝐹 “ {𝑍})) = ran (𝐹 ↾ (𝐹 “ {𝑍}))
44 funimacnv 6305 . . . . . . 7 (Fun 𝐹 → (𝐹 “ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
4512, 44syl 17 . . . . . 6 (𝜑 → (𝐹 “ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
4643, 45syl5eqr 2845 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
47 inss1 4125 . . . . 5 ({𝑍} ∩ ran 𝐹) ⊆ {𝑍}
4846, 47syl6eqss 3942 . . . 4 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) ⊆ {𝑍})
49 ssfi 8584 . . . 4 (({𝑍} ∈ Fin ∧ ran (𝐹 ↾ (𝐹 “ {𝑍})) ⊆ {𝑍}) → ran (𝐹 ↾ (𝐹 “ {𝑍})) ∈ Fin)
5042, 48, 49sylancr 587 . . 3 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) ∈ Fin)
51 unfi 8631 . . 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 584 . 2 (𝜑 → (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍}))) ∈ Fin)
5323, 52eqeltrd 2883 1 (𝜑 → ran 𝐹 ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ∖ cdif 3856   ∪ cun 3857   ∩ cin 3858   ⊆ wss 3859  {csn 4472   class class class wbr 4962  ◡ccnv 5442  dom cdm 5443  ran crn 5444   ↾ cres 5445   “ cima 5446  Fun wfun 6219   Fn wfn 6220  –onto→wfo 6223  (class class class)co 7016   supp csupp 7681   ≼ cdom 8355  Fincfn 8357 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-ac2 9731 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-supp 7682  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-fin 8361  df-card 9214  df-acn 9217  df-ac 9388 This theorem is referenced by:  fpwrelmapffslem  30156
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