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Theorem indf1ofs 31994
Description: The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)
Assertion
Ref Expression
indf1ofs (𝑂𝑉 → ((𝟭‘𝑂) ↾ Fin):(𝒫 𝑂 ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (𝑓 “ {1}) ∈ Fin})
Distinct variable group:   𝑓,𝑂
Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem indf1ofs
Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 indf1o 31992 . . . 4 (𝑂𝑉 → (𝟭‘𝑂):𝒫 𝑂1-1-onto→({0, 1} ↑m 𝑂))
2 f1of1 6715 . . . 4 ((𝟭‘𝑂):𝒫 𝑂1-1-onto→({0, 1} ↑m 𝑂) → (𝟭‘𝑂):𝒫 𝑂1-1→({0, 1} ↑m 𝑂))
31, 2syl 17 . . 3 (𝑂𝑉 → (𝟭‘𝑂):𝒫 𝑂1-1→({0, 1} ↑m 𝑂))
4 inss1 4162 . . 3 (𝒫 𝑂 ∩ Fin) ⊆ 𝒫 𝑂
5 f1ores 6730 . . 3 (((𝟭‘𝑂):𝒫 𝑂1-1→({0, 1} ↑m 𝑂) ∧ (𝒫 𝑂 ∩ Fin) ⊆ 𝒫 𝑂) → ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)):(𝒫 𝑂 ∩ Fin)–1-1-onto→((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)))
63, 4, 5sylancl 586 . 2 (𝑂𝑉 → ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)):(𝒫 𝑂 ∩ Fin)–1-1-onto→((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)))
7 resres 5904 . . . 4 (((𝟭‘𝑂) ↾ 𝒫 𝑂) ↾ Fin) = ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin))
8 f1ofn 6717 . . . . . 6 ((𝟭‘𝑂):𝒫 𝑂1-1-onto→({0, 1} ↑m 𝑂) → (𝟭‘𝑂) Fn 𝒫 𝑂)
9 fnresdm 6551 . . . . . 6 ((𝟭‘𝑂) Fn 𝒫 𝑂 → ((𝟭‘𝑂) ↾ 𝒫 𝑂) = (𝟭‘𝑂))
101, 8, 93syl 18 . . . . 5 (𝑂𝑉 → ((𝟭‘𝑂) ↾ 𝒫 𝑂) = (𝟭‘𝑂))
1110reseq1d 5890 . . . 4 (𝑂𝑉 → (((𝟭‘𝑂) ↾ 𝒫 𝑂) ↾ Fin) = ((𝟭‘𝑂) ↾ Fin))
127, 11eqtr3id 2792 . . 3 (𝑂𝑉 → ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)) = ((𝟭‘𝑂) ↾ Fin))
13 eqidd 2739 . . 3 (𝑂𝑉 → (𝒫 𝑂 ∩ Fin) = (𝒫 𝑂 ∩ Fin))
14 simpll 764 . . . . . . . . 9 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑂𝑉)
15 simpr 485 . . . . . . . . . . . . . 14 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ (𝒫 𝑂 ∩ Fin))
164, 15sselid 3919 . . . . . . . . . . . . 13 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ 𝒫 𝑂)
1716elpwid 4544 . . . . . . . . . . . 12 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎𝑂)
18 indf 31983 . . . . . . . . . . . 12 ((𝑂𝑉𝑎𝑂) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
1917, 18syldan 591 . . . . . . . . . . 11 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
2019adantr 481 . . . . . . . . . 10 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
21 simpr 485 . . . . . . . . . . 11 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ((𝟭‘𝑂)‘𝑎) = 𝑔)
2221feq1d 6585 . . . . . . . . . 10 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1} ↔ 𝑔:𝑂⟶{0, 1}))
2320, 22mpbid 231 . . . . . . . . 9 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑔:𝑂⟶{0, 1})
24 prex 5355 . . . . . . . . . . 11 {0, 1} ∈ V
25 elmapg 8628 . . . . . . . . . . 11 (({0, 1} ∈ V ∧ 𝑂𝑉) → (𝑔 ∈ ({0, 1} ↑m 𝑂) ↔ 𝑔:𝑂⟶{0, 1}))
2624, 25mpan 687 . . . . . . . . . 10 (𝑂𝑉 → (𝑔 ∈ ({0, 1} ↑m 𝑂) ↔ 𝑔:𝑂⟶{0, 1}))
2726biimpar 478 . . . . . . . . 9 ((𝑂𝑉𝑔:𝑂⟶{0, 1}) → 𝑔 ∈ ({0, 1} ↑m 𝑂))
2814, 23, 27syl2anc 584 . . . . . . . 8 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑔 ∈ ({0, 1} ↑m 𝑂))
2921cnveqd 5784 . . . . . . . . . 10 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ((𝟭‘𝑂)‘𝑎) = 𝑔)
3029imaeq1d 5968 . . . . . . . . 9 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (((𝟭‘𝑂)‘𝑎) “ {1}) = (𝑔 “ {1}))
31 indpi1 31988 . . . . . . . . . . . 12 ((𝑂𝑉𝑎𝑂) → (((𝟭‘𝑂)‘𝑎) “ {1}) = 𝑎)
3217, 31syldan 591 . . . . . . . . . . 11 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → (((𝟭‘𝑂)‘𝑎) “ {1}) = 𝑎)
33 inss2 4163 . . . . . . . . . . . 12 (𝒫 𝑂 ∩ Fin) ⊆ Fin
3433, 15sselid 3919 . . . . . . . . . . 11 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ Fin)
3532, 34eqeltrd 2839 . . . . . . . . . 10 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → (((𝟭‘𝑂)‘𝑎) “ {1}) ∈ Fin)
3635adantr 481 . . . . . . . . 9 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (((𝟭‘𝑂)‘𝑎) “ {1}) ∈ Fin)
3730, 36eqeltrrd 2840 . . . . . . . 8 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (𝑔 “ {1}) ∈ Fin)
3828, 37jca 512 . . . . . . 7 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin))
3938rexlimdva2 3216 . . . . . 6 (𝑂𝑉 → (∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔 → (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)))
40 cnvimass 5989 . . . . . . . . . 10 (𝑔 “ {1}) ⊆ dom 𝑔
4126biimpa 477 . . . . . . . . . . . 12 ((𝑂𝑉𝑔 ∈ ({0, 1} ↑m 𝑂)) → 𝑔:𝑂⟶{0, 1})
4241fdmd 6611 . . . . . . . . . . 11 ((𝑂𝑉𝑔 ∈ ({0, 1} ↑m 𝑂)) → dom 𝑔 = 𝑂)
4342adantrr 714 . . . . . . . . . 10 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → dom 𝑔 = 𝑂)
4440, 43sseqtrid 3973 . . . . . . . . 9 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → (𝑔 “ {1}) ⊆ 𝑂)
45 simprr 770 . . . . . . . . 9 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → (𝑔 “ {1}) ∈ Fin)
46 elfpw 9121 . . . . . . . . 9 ((𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin) ↔ ((𝑔 “ {1}) ⊆ 𝑂 ∧ (𝑔 “ {1}) ∈ Fin))
4744, 45, 46sylanbrc 583 . . . . . . . 8 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → (𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin))
48 indpreima 31993 . . . . . . . . . . 11 ((𝑂𝑉𝑔:𝑂⟶{0, 1}) → 𝑔 = ((𝟭‘𝑂)‘(𝑔 “ {1})))
4948eqcomd 2744 . . . . . . . . . 10 ((𝑂𝑉𝑔:𝑂⟶{0, 1}) → ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔)
5041, 49syldan 591 . . . . . . . . 9 ((𝑂𝑉𝑔 ∈ ({0, 1} ↑m 𝑂)) → ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔)
5150adantrr 714 . . . . . . . 8 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔)
52 fveqeq2 6783 . . . . . . . . 9 (𝑎 = (𝑔 “ {1}) → (((𝟭‘𝑂)‘𝑎) = 𝑔 ↔ ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔))
5352rspcev 3561 . . . . . . . 8 (((𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin) ∧ ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔)
5447, 51, 53syl2anc 584 . . . . . . 7 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔)
5554ex 413 . . . . . 6 (𝑂𝑉 → ((𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔))
5639, 55impbid 211 . . . . 5 (𝑂𝑉 → (∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔 ↔ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)))
571, 8syl 17 . . . . . 6 (𝑂𝑉 → (𝟭‘𝑂) Fn 𝒫 𝑂)
58 fvelimab 6841 . . . . . 6 (((𝟭‘𝑂) Fn 𝒫 𝑂 ∧ (𝒫 𝑂 ∩ Fin) ⊆ 𝒫 𝑂) → (𝑔 ∈ ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔))
5957, 4, 58sylancl 586 . . . . 5 (𝑂𝑉 → (𝑔 ∈ ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔))
60 cnveq 5782 . . . . . . . . 9 (𝑓 = 𝑔𝑓 = 𝑔)
6160imaeq1d 5968 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓 “ {1}) = (𝑔 “ {1}))
6261eleq1d 2823 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓 “ {1}) ∈ Fin ↔ (𝑔 “ {1}) ∈ Fin))
6362elrab 3624 . . . . . 6 (𝑔 ∈ {𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (𝑓 “ {1}) ∈ Fin} ↔ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin))
6463a1i 11 . . . . 5 (𝑂𝑉 → (𝑔 ∈ {𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (𝑓 “ {1}) ∈ Fin} ↔ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)))
6556, 59, 643bitr4d 311 . . . 4 (𝑂𝑉 → (𝑔 ∈ ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ 𝑔 ∈ {𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (𝑓 “ {1}) ∈ Fin}))
6665eqrdv 2736 . . 3 (𝑂𝑉 → ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) = {𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (𝑓 “ {1}) ∈ Fin})
6712, 13, 66f1oeq123d 6710 . 2 (𝑂𝑉 → (((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)):(𝒫 𝑂 ∩ Fin)–1-1-onto→((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ ((𝟭‘𝑂) ↾ Fin):(𝒫 𝑂 ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (𝑓 “ {1}) ∈ Fin}))
686, 67mpbid 231 1 (𝑂𝑉 → ((𝟭‘𝑂) ↾ Fin):(𝒫 𝑂 ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (𝑓 “ {1}) ∈ Fin})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wrex 3065  {crab 3068  Vcvv 3432  cin 3886  wss 3887  𝒫 cpw 4533  {csn 4561  {cpr 4563  ccnv 5588  dom cdm 5589  cres 5591  cima 5592   Fn wfn 6428  wf 6429  1-1wf1 6430  1-1-ontowf1o 6432  cfv 6433  (class class class)co 7275  m cmap 8615  Fincfn 8733  0cc0 10871  1c1 10872  𝟭cind 31978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-i2m1 10939  ax-1ne0 10940  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-ind 31979
This theorem is referenced by:  eulerpartgbij  32339
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