Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ffsrn Structured version   Visualization version   GIF version

Theorem ffsrn 32423
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 ffsrn.1 . . . . . 6 (𝜑 → Fun 𝐹)
2 dfdm4 5885 . . . . . . 7 dom 𝐹 = ran 𝐹
3 dfrn4 6191 . . . . . . 7 ran 𝐹 = (𝐹 “ V)
42, 3eqtri 2752 . . . . . 6 dom 𝐹 = (𝐹 “ V)
5 df-fn 6536 . . . . . . 7 (𝐹 Fn (𝐹 “ V) ↔ (Fun 𝐹 ∧ dom 𝐹 = (𝐹 “ V)))
6 fnresdm 6659 . . . . . . 7 (𝐹 Fn (𝐹 “ V) → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
75, 6sylbir 234 . . . . . 6 ((Fun 𝐹 ∧ dom 𝐹 = (𝐹 “ V)) → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
81, 4, 7sylancl 585 . . . . 5 (𝜑 → (𝐹 ↾ (𝐹 “ V)) = 𝐹)
9 imaundi 6139 . . . . . . 7 (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍}))
109reseq2i 5968 . . . . . 6 (𝐹 ↾ (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍})))
11 undif1 4467 . . . . . . . . 9 ((V ∖ {𝑍}) ∪ {𝑍}) = (V ∪ {𝑍})
12 ssv 3998 . . . . . . . . . 10 {𝑍} ⊆ V
13 ssequn2 4175 . . . . . . . . . 10 ({𝑍} ⊆ V ↔ (V ∪ {𝑍}) = V)
1412, 13mpbi 229 . . . . . . . . 9 (V ∪ {𝑍}) = V
1511, 14eqtri 2752 . . . . . . . 8 ((V ∖ {𝑍}) ∪ {𝑍}) = V
1615imaeq2i 6047 . . . . . . 7 (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍})) = (𝐹 “ V)
1716reseq2i 5968 . . . . . 6 (𝐹 ↾ (𝐹 “ ((V ∖ {𝑍}) ∪ {𝑍}))) = (𝐹 ↾ (𝐹 “ V))
18 resundi 5985 . . . . . 6 (𝐹 ↾ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐹 “ {𝑍}))) = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍})))
1910, 17, 183eqtr3i 2760 . . . . 5 (𝐹 ↾ (𝐹 “ V)) = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍})))
208, 19eqtr3di 2779 . . . 4 (𝜑𝐹 = ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))))
2120rneqd 5927 . . 3 (𝜑 → ran 𝐹 = ran ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))))
22 rnun 6135 . . 3 ran ((𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ (𝐹 ↾ (𝐹 “ {𝑍}))) = (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍})))
2321, 22eqtrdi 2780 . 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 8153 . . . . . 6 ((𝐹𝑉𝑍𝑊) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
2724, 25, 26syl2anc 583 . . . . 5 (𝜑 → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
28 ffsrn.2 . . . . 5 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
2927, 28eqeltrrd 2826 . . . 4 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ Fin)
30 cnvexg 7908 . . . . . 6 (𝐹𝑉𝐹 ∈ V)
31 imaexg 7899 . . . . . 6 (𝐹 ∈ V → (𝐹 “ (V ∖ {𝑍})) ∈ V)
3224, 30, 313syl 18 . . . . 5 (𝜑 → (𝐹 “ (V ∖ {𝑍})) ∈ V)
33 cnvimass 6070 . . . . . . 7 (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹
34 fores 6805 . . . . . . 7 ((Fun 𝐹 ∧ (𝐹 “ (V ∖ {𝑍})) ⊆ dom 𝐹) → (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))):(𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (𝐹 “ (V ∖ {𝑍}))))
351, 33, 34sylancl 585 . . . . . 6 (𝜑 → (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))):(𝐹 “ (V ∖ {𝑍}))–onto→(𝐹 “ (𝐹 “ (V ∖ {𝑍}))))
36 fofn 6797 . . . . . 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 10527 . . . . 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 9188 . . . 4 (((𝐹 “ (V ∖ {𝑍})) ∈ Fin ∧ ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ≼ (𝐹 “ (V ∖ {𝑍}))) → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
4129, 39, 40syl2anc 583 . . 3 (𝜑 → ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∈ Fin)
42 snfi 9040 . . . 4 {𝑍} ∈ Fin
43 df-ima 5679 . . . . . 6 (𝐹 “ (𝐹 “ {𝑍})) = ran (𝐹 ↾ (𝐹 “ {𝑍}))
44 funimacnv 6619 . . . . . . 7 (Fun 𝐹 → (𝐹 “ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
451, 44syl 17 . . . . . 6 (𝜑 → (𝐹 “ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
4643, 45eqtr3id 2778 . . . . 5 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) = ({𝑍} ∩ ran 𝐹))
47 inss1 4220 . . . . 5 ({𝑍} ∩ ran 𝐹) ⊆ {𝑍}
4846, 47eqsstrdi 4028 . . . 4 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) ⊆ {𝑍})
49 ssfi 9169 . . . 4 (({𝑍} ∈ Fin ∧ ran (𝐹 ↾ (𝐹 “ {𝑍})) ⊆ {𝑍}) → ran (𝐹 ↾ (𝐹 “ {𝑍})) ∈ Fin)
5042, 48, 49sylancr 586 . . 3 (𝜑 → ran (𝐹 ↾ (𝐹 “ {𝑍})) ∈ Fin)
51 unfi 9168 . . 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 583 . 2 (𝜑 → (ran (𝐹 ↾ (𝐹 “ (V ∖ {𝑍}))) ∪ ran (𝐹 ↾ (𝐹 “ {𝑍}))) ∈ Fin)
5323, 52eqeltrd 2825 1 (𝜑 → ran 𝐹 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  cdif 3937  cun 3938  cin 3939  wss 3940  {csn 4620   class class class wbr 5138  ccnv 5665  dom cdm 5666  ran crn 5667  cres 5668  cima 5669  Fun wfun 6527   Fn wfn 6528  ontowfo 6531  (class class class)co 7401   supp csupp 8140  cdom 8933  Fincfn 8935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-ac2 10454
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-1o 8461  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-fin 8939  df-card 9930  df-acn 9933  df-ac 10107
This theorem is referenced by:  fpwrelmapffslem  32426
  Copyright terms: Public domain W3C validator