Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  indf1ofs Structured version   Visualization version   GIF version

Theorem indf1ofs 31280
 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 31278 . . . 4 (𝑂𝑉 → (𝟭‘𝑂):𝒫 𝑂1-1-onto→({0, 1} ↑m 𝑂))
2 f1of1 6608 . . . 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 4204 . . 3 (𝒫 𝑂 ∩ Fin) ⊆ 𝒫 𝑂
5 f1ores 6623 . . 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 588 . 2 (𝑂𝑉 → ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)):(𝒫 𝑂 ∩ Fin)–1-1-onto→((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)))
7 resres 5860 . . . 4 (((𝟭‘𝑂) ↾ 𝒫 𝑂) ↾ Fin) = ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin))
8 f1ofn 6610 . . . . . 6 ((𝟭‘𝑂):𝒫 𝑂1-1-onto→({0, 1} ↑m 𝑂) → (𝟭‘𝑂) Fn 𝒫 𝑂)
9 fnresdm 6460 . . . . . 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 5846 . . . 4 (𝑂𝑉 → (((𝟭‘𝑂) ↾ 𝒫 𝑂) ↾ Fin) = ((𝟭‘𝑂) ↾ Fin))
127, 11syl5eqr 2870 . . 3 (𝑂𝑉 → ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)) = ((𝟭‘𝑂) ↾ Fin))
13 eqidd 2822 . . 3 (𝑂𝑉 → (𝒫 𝑂 ∩ Fin) = (𝒫 𝑂 ∩ Fin))
14 simpll 765 . . . . . . . . 9 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑂𝑉)
15 simpr 487 . . . . . . . . . . . . . 14 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ (𝒫 𝑂 ∩ Fin))
164, 15sseldi 3964 . . . . . . . . . . . . 13 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ 𝒫 𝑂)
1716elpwid 4552 . . . . . . . . . . . 12 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎𝑂)
18 indf 31269 . . . . . . . . . . . 12 ((𝑂𝑉𝑎𝑂) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
1917, 18syldan 593 . . . . . . . . . . 11 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
2019adantr 483 . . . . . . . . . 10 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
21 simpr 487 . . . . . . . . . . 11 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ((𝟭‘𝑂)‘𝑎) = 𝑔)
2221feq1d 6493 . . . . . . . . . 10 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1} ↔ 𝑔:𝑂⟶{0, 1}))
2320, 22mpbid 234 . . . . . . . . 9 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑔:𝑂⟶{0, 1})
24 prex 5324 . . . . . . . . . . 11 {0, 1} ∈ V
25 elmapg 8413 . . . . . . . . . . 11 (({0, 1} ∈ V ∧ 𝑂𝑉) → (𝑔 ∈ ({0, 1} ↑m 𝑂) ↔ 𝑔:𝑂⟶{0, 1}))
2624, 25mpan 688 . . . . . . . . . 10 (𝑂𝑉 → (𝑔 ∈ ({0, 1} ↑m 𝑂) ↔ 𝑔:𝑂⟶{0, 1}))
2726biimpar 480 . . . . . . . . 9 ((𝑂𝑉𝑔:𝑂⟶{0, 1}) → 𝑔 ∈ ({0, 1} ↑m 𝑂))
2814, 23, 27syl2anc 586 . . . . . . . 8 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑔 ∈ ({0, 1} ↑m 𝑂))
2921cnveqd 5740 . . . . . . . . . 10 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ((𝟭‘𝑂)‘𝑎) = 𝑔)
3029imaeq1d 5922 . . . . . . . . 9 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (((𝟭‘𝑂)‘𝑎) “ {1}) = (𝑔 “ {1}))
31 indpi1 31274 . . . . . . . . . . . 12 ((𝑂𝑉𝑎𝑂) → (((𝟭‘𝑂)‘𝑎) “ {1}) = 𝑎)
3217, 31syldan 593 . . . . . . . . . . 11 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → (((𝟭‘𝑂)‘𝑎) “ {1}) = 𝑎)
33 inss2 4205 . . . . . . . . . . . 12 (𝒫 𝑂 ∩ Fin) ⊆ Fin
3433, 15sseldi 3964 . . . . . . . . . . 11 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ Fin)
3532, 34eqeltrd 2913 . . . . . . . . . 10 ((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → (((𝟭‘𝑂)‘𝑎) “ {1}) ∈ Fin)
3635adantr 483 . . . . . . . . 9 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (((𝟭‘𝑂)‘𝑎) “ {1}) ∈ Fin)
3730, 36eqeltrrd 2914 . . . . . . . 8 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (𝑔 “ {1}) ∈ Fin)
3828, 37jca 514 . . . . . . 7 (((𝑂𝑉𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin))
3938rexlimdva2 3287 . . . . . 6 (𝑂𝑉 → (∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔 → (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)))
40 cnvimass 5943 . . . . . . . . . 10 (𝑔 “ {1}) ⊆ dom 𝑔
4126biimpa 479 . . . . . . . . . . . 12 ((𝑂𝑉𝑔 ∈ ({0, 1} ↑m 𝑂)) → 𝑔:𝑂⟶{0, 1})
4241fdmd 6517 . . . . . . . . . . 11 ((𝑂𝑉𝑔 ∈ ({0, 1} ↑m 𝑂)) → dom 𝑔 = 𝑂)
4342adantrr 715 . . . . . . . . . 10 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → dom 𝑔 = 𝑂)
4440, 43sseqtrid 4018 . . . . . . . . 9 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → (𝑔 “ {1}) ⊆ 𝑂)
45 simprr 771 . . . . . . . . 9 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → (𝑔 “ {1}) ∈ Fin)
46 elfpw 8820 . . . . . . . . 9 ((𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin) ↔ ((𝑔 “ {1}) ⊆ 𝑂 ∧ (𝑔 “ {1}) ∈ Fin))
4744, 45, 46sylanbrc 585 . . . . . . . 8 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → (𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin))
48 indpreima 31279 . . . . . . . . . . 11 ((𝑂𝑉𝑔:𝑂⟶{0, 1}) → 𝑔 = ((𝟭‘𝑂)‘(𝑔 “ {1})))
4948eqcomd 2827 . . . . . . . . . 10 ((𝑂𝑉𝑔:𝑂⟶{0, 1}) → ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔)
5041, 49syldan 593 . . . . . . . . 9 ((𝑂𝑉𝑔 ∈ ({0, 1} ↑m 𝑂)) → ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔)
5150adantrr 715 . . . . . . . 8 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔)
52 fveqeq2 6673 . . . . . . . . 9 (𝑎 = (𝑔 “ {1}) → (((𝟭‘𝑂)‘𝑎) = 𝑔 ↔ ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔))
5352rspcev 3622 . . . . . . . 8 (((𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin) ∧ ((𝟭‘𝑂)‘(𝑔 “ {1})) = 𝑔) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔)
5447, 51, 53syl2anc 586 . . . . . . 7 ((𝑂𝑉 ∧ (𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin)) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔)
5554ex 415 . . . . . 6 (𝑂𝑉 → ((𝑔 ∈ ({0, 1} ↑m 𝑂) ∧ (𝑔 “ {1}) ∈ Fin) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔))
5639, 55impbid 214 . . . . 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 6731 . . . . . 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 588 . . . . 5 (𝑂𝑉 → (𝑔 ∈ ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔))
60 cnveq 5738 . . . . . . . . 9 (𝑓 = 𝑔𝑓 = 𝑔)
6160imaeq1d 5922 . . . . . . . 8 (𝑓 = 𝑔 → (𝑓 “ {1}) = (𝑔 “ {1}))
6261eleq1d 2897 . . . . . . 7 (𝑓 = 𝑔 → ((𝑓 “ {1}) ∈ Fin ↔ (𝑔 “ {1}) ∈ Fin))
6362elrab 3679 . . . . . 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 313 . . . 4 (𝑂𝑉 → (𝑔 ∈ ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ 𝑔 ∈ {𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (𝑓 “ {1}) ∈ Fin}))
6665eqrdv 2819 . . 3 (𝑂𝑉 → ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) = {𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (𝑓 “ {1}) ∈ Fin})
6712, 13, 66f1oeq123d 6604 . 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 234 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 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  {crab 3142  Vcvv 3494   ∩ cin 3934   ⊆ wss 3935  𝒫 cpw 4538  {csn 4560  {cpr 4562  ◡ccnv 5548  dom cdm 5549   ↾ cres 5551   “ cima 5552   Fn wfn 6344  ⟶wf 6345  –1-1→wf1 6346  –1-1-onto→wf1o 6348  ‘cfv 6349  (class class class)co 7150   ↑m cmap 8400  Fincfn 8503  0cc0 10531  1c1 10532  𝟭cind 31264 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-i2m1 10599  ax-1ne0 10600  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8402  df-ind 31265 This theorem is referenced by:  eulerpartgbij  31625
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