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Theorem indf1ofs 30062
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 30060 . . . 4 (𝑂𝑉 → (𝟭‘𝑂):𝒫 𝑂1-1-onto→({0, 1} ↑𝑚 𝑂))
2 f1of1 6123 . . . 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 3825 . . 3 (𝒫 𝑂 ∩ Fin) ⊆ 𝒫 𝑂
5 f1ores 6138 . . 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 693 . 2 (𝑂𝑉 → ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)):(𝒫 𝑂 ∩ Fin)–1-1-onto→((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)))
7 resres 5397 . . . 4 (((𝟭‘𝑂) ↾ 𝒫 𝑂) ↾ Fin) = ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin))
8 f1ofn 6125 . . . . . 6 ((𝟭‘𝑂):𝒫 𝑂1-1-onto→({0, 1} ↑𝑚 𝑂) → (𝟭‘𝑂) Fn 𝒫 𝑂)
9 fnresdm 5988 . . . . . 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 5384 . . . 4 (𝑂𝑉 → (((𝟭‘𝑂) ↾ 𝒫 𝑂) ↾ Fin) = ((𝟭‘𝑂) ↾ Fin))
127, 11syl5eqr 2668 . . 3 (𝑂𝑉 → ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)) = ((𝟭‘𝑂) ↾ Fin))
13 eqidd 2621 . . 3 (𝑂𝑉 → (𝒫 𝑂 ∩ Fin) = (𝒫 𝑂 ∩ Fin))
14 simpll 789 . . . . . . . . . 10 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑂𝑉)
15 simpr 477 . . . . . . . . . . . . . . 15 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ (𝒫 𝑂 ∩ Fin))
164, 15sseldi 3593 . . . . . . . . . . . . . 14 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ 𝒫 𝑂)
1716elpwid 4161 . . . . . . . . . . . . 13 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎𝑂)
18 indf 30051 . . . . . . . . . . . . 13 ((𝑂𝑉𝑎𝑂) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
1917, 18syldan 487 . . . . . . . . . . . 12 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
2019adantr 481 . . . . . . . . . . 11 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
21 simpr 477 . . . . . . . . . . . 12 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ((𝟭‘𝑂)‘𝑎) = 𝑔)
2221feq1d 6017 . . . . . . . . . . 11 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1} ↔ 𝑔:𝑂⟶{0, 1}))
2320, 22mpbid 222 . . . . . . . . . 10 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑔:𝑂⟶{0, 1})
24 prex 4900 . . . . . . . . . . . 12 {0, 1} ∈ V
25 elmapg 7855 . . . . . . . . . . . 12 (({0, 1} ∈ V ∧ 𝑂𝑉) → (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ↔ 𝑔:𝑂⟶{0, 1}))
2624, 25mpan 705 . . . . . . . . . . 11 (𝑂𝑉 → (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ↔ 𝑔:𝑂⟶{0, 1}))
2726biimpar 502 . . . . . . . . . 10 ((𝑂𝑉𝑔:𝑂⟶{0, 1}) → 𝑔 ∈ ({0, 1} ↑𝑚 𝑂))
2814, 23, 27syl2anc 692 . . . . . . . . 9 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑔 ∈ ({0, 1} ↑𝑚 𝑂))
2921cnveqd 5287 . . . . . . . . . . 11 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ((𝟭‘𝑂)‘𝑎) = 𝑔)
3029imaeq1d 5453 . . . . . . . . . 10 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (((𝟭‘𝑂)‘𝑎) “ {1}) = (𝑔 “ {1}))
31 indpi1 30056 . . . . . . . . . . . . 13 ((𝑂𝑉𝑎𝑂) → (((𝟭‘𝑂)‘𝑎) “ {1}) = 𝑎)
3217, 31syldan 487 . . . . . . . . . . . 12 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → (((𝟭‘𝑂)‘𝑎) “ {1}) = 𝑎)
33 inss2 3826 . . . . . . . . . . . . 13 (𝒫 𝑂 ∩ Fin) ⊆ Fin
3433, 15sseldi 3593 . . . . . . . . . . . 12 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ Fin)
3532, 34eqeltrd 2699 . . . . . . . . . . 11 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → (((𝟭‘𝑂)‘𝑎) “ {1}) ∈ Fin)
3635adantr 481 . . . . . . . . . 10 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (((𝟭‘𝑂)‘𝑎) “ {1}) ∈ Fin)
3730, 36eqeltrrd 2700 . . . . . . . . 9 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (𝑔 “ {1}) ∈ Fin)
3828, 37jca 554 . . . . . . . 8 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin))
3938ex 450 . . . . . . 7 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → (((𝟭‘𝑂)‘𝑎) = 𝑔 → (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)))
4039rexlimdva 3027 . . . . . 6 (𝑂𝑉 → (∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔 → (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)))
41 cnvimass 5473 . . . . . . . . . 10 (𝑔 “ {1}) ⊆ dom 𝑔
4226biimpa 501 . . . . . . . . . . . 12 ((𝑂𝑉𝑔 ∈ ({0, 1} ↑𝑚 𝑂)) → 𝑔:𝑂⟶{0, 1})
43 fdm 6038 . . . . . . . . . . . 12 (𝑔:𝑂⟶{0, 1} → dom 𝑔 = 𝑂)
4442, 43syl 17 . . . . . . . . . . 11 ((𝑂𝑉𝑔 ∈ ({0, 1} ↑𝑚 𝑂)) → dom 𝑔 = 𝑂)
4544adantrr 752 . . . . . . . . . 10 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → dom 𝑔 = 𝑂)
4641, 45syl5sseq 3645 . . . . . . . . 9 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → (𝑔 “ {1}) ⊆ 𝑂)
47 simprr 795 . . . . . . . . 9 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → (𝑔 “ {1}) ∈ Fin)
48 elfpw 8253 . . . . . . . . 9 ((𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin) ↔ ((𝑔 “ {1}) ⊆ 𝑂 ∧ (𝑔 “ {1}) ∈ Fin))
4946, 47, 48sylanbrc 697 . . . . . . . 8 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → (𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin))
50 indpreima 30061 . . . . . . . . . . 11 ((𝑂𝑉𝑔:𝑂⟶{0, 1}) → 𝑔 = ((𝟭‘𝑂)‘(𝑔 “ {1})))
5150eqcomd 2626 . . . . . . . . . 10 ((𝑂𝑉𝑔:𝑂⟶{0, 1}) → ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔)
5242, 51syldan 487 . . . . . . . . 9 ((𝑂𝑉𝑔 ∈ ({0, 1} ↑𝑚 𝑂)) → ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔)
5352adantrr 752 . . . . . . . 8 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔)
54 fveq2 6178 . . . . . . . . . 10 (𝑎 = (𝑔 “ {1}) → ((𝟭‘𝑂)‘𝑎) = ((𝟭‘𝑂)‘(𝑔 “ {1})))
5554eqeq1d 2622 . . . . . . . . 9 (𝑎 = (𝑔 “ {1}) → (((𝟭‘𝑂)‘𝑎) = 𝑔 ↔ ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔))
5655rspcev 3304 . . . . . . . 8 (((𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin) ∧ ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔)
5749, 53, 56syl2anc 692 . . . . . . 7 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔)
5857ex 450 . . . . . 6 (𝑂𝑉 → ((𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔))
5940, 58impbid 202 . . . . 5 (𝑂𝑉 → (∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔 ↔ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)))
601, 8syl 17 . . . . . 6 (𝑂𝑉 → (𝟭‘𝑂) Fn 𝒫 𝑂)
61 fvelimab 6240 . . . . . 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))
6260, 4, 61sylancl 693 . . . . 5 (𝑂𝑉 → (𝑔 ∈ ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔))
63 cnveq 5285 . . . . . . . . 9 (𝑓 = 𝑔𝑓 = 𝑔)
6463imaeq1d 5453 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓 “ {1}) = (𝑔 “ {1}))
6564eleq1d 2684 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓 “ {1}) ∈ Fin ↔ (𝑔 “ {1}) ∈ Fin))
6665elrab 3357 . . . . . 6 (𝑔 ∈ {𝑓 ∈ ({0, 1} ↑𝑚 𝑂) ∣ (𝑓 “ {1}) ∈ Fin} ↔ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin))
6766a1i 11 . . . . 5 (𝑂𝑉 → (𝑔 ∈ {𝑓 ∈ ({0, 1} ↑𝑚 𝑂) ∣ (𝑓 “ {1}) ∈ Fin} ↔ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)))
6859, 62, 673bitr4d 300 . . . 4 (𝑂𝑉 → (𝑔 ∈ ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ 𝑔 ∈ {𝑓 ∈ ({0, 1} ↑𝑚 𝑂) ∣ (𝑓 “ {1}) ∈ Fin}))
6968eqrdv 2618 . . 3 (𝑂𝑉 → ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) = {𝑓 ∈ ({0, 1} ↑𝑚 𝑂) ∣ (𝑓 “ {1}) ∈ Fin})
7012, 13, 69f1oeq123d 6120 . 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}))
716, 70mpbid 222 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 196  wa 384   = wceq 1481  wcel 1988  wrex 2910  {crab 2913  Vcvv 3195  cin 3566  wss 3567  𝒫 cpw 4149  {csn 4168  {cpr 4170  ccnv 5103  dom cdm 5104  cres 5106  cima 5107   Fn wfn 5871  wf 5872  1-1wf1 5873  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  𝑚 cmap 7842  Fincfn 7940  0cc0 9921  1c1 9922  𝟭cind 30046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-i2m1 9989  ax-1ne0 9990  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-ind 30047
This theorem is referenced by:  eulerpartgbij  30408
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