Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwfi2f1o Structured version   Visualization version   GIF version

Theorem pwfi2f1o 39574
Description: The pw2f1o 8610 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.)
Hypotheses
Ref Expression
pwfi2f1o.s 𝑆 = {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅}
pwfi2f1o.f 𝐹 = (𝑥𝑆 ↦ (𝑥 “ {1o}))
Assertion
Ref Expression
pwfi2f1o (𝐴𝑉𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑆   𝑥,𝑉,𝑦
Allowed substitution hints:   𝑆(𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem pwfi2f1o
StepHypRef Expression
1 eqid 2818 . . . . 5 (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) = (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o}))
21pw2f1o2 39513 . . . 4 (𝐴𝑉 → (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–1-1-onto→𝒫 𝐴)
3 f1of1 6607 . . . 4 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–1-1-onto→𝒫 𝐴 → (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–1-1→𝒫 𝐴)
42, 3syl 17 . . 3 (𝐴𝑉 → (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–1-1→𝒫 𝐴)
5 pwfi2f1o.s . . . 4 𝑆 = {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅}
6 ssrab2 4053 . . . 4 {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2om 𝐴)
75, 6eqsstri 3998 . . 3 𝑆 ⊆ (2om 𝐴)
8 f1ores 6622 . . 3 (((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–1-1→𝒫 𝐴𝑆 ⊆ (2om 𝐴)) → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆))
94, 7, 8sylancl 586 . 2 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆))
10 elmapfun 8419 . . . . . . . . . . . . 13 (𝑦 ∈ (2om 𝐴) → Fun 𝑦)
11 id 22 . . . . . . . . . . . . 13 (𝑦 ∈ (2om 𝐴) → 𝑦 ∈ (2om 𝐴))
12 0ex 5202 . . . . . . . . . . . . . 14 ∅ ∈ V
1312a1i 11 . . . . . . . . . . . . 13 (𝑦 ∈ (2om 𝐴) → ∅ ∈ V)
1410, 11, 133jca 1120 . . . . . . . . . . . 12 (𝑦 ∈ (2om 𝐴) → (Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V))
1514adantl 482 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V))
16 funisfsupp 8826 . . . . . . . . . . 11 ((Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1715, 16syl 17 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1813anim2i 616 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝐴𝑉 ∧ ∅ ∈ V))
19 elmapi 8417 . . . . . . . . . . . . . 14 (𝑦 ∈ (2om 𝐴) → 𝑦:𝐴⟶2o)
2019adantl 482 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → 𝑦:𝐴⟶2o)
21 frnsuppeq 7831 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ ∅ ∈ V) → (𝑦:𝐴⟶2o → (𝑦 supp ∅) = (𝑦 “ (2o ∖ {∅}))))
2218, 20, 21sylc 65 . . . . . . . . . . . 12 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 supp ∅) = (𝑦 “ (2o ∖ {∅})))
23 df-2o 8092 . . . . . . . . . . . . . . . 16 2o = suc 1o
24 df-suc 6190 . . . . . . . . . . . . . . . . 17 suc 1o = (1o ∪ {1o})
2524equncomi 4128 . . . . . . . . . . . . . . . 16 suc 1o = ({1o} ∪ 1o)
2623, 25eqtri 2841 . . . . . . . . . . . . . . 15 2o = ({1o} ∪ 1o)
27 df1o2 8105 . . . . . . . . . . . . . . . 16 1o = {∅}
2827eqcomi 2827 . . . . . . . . . . . . . . 15 {∅} = 1o
2926, 28difeq12i 4094 . . . . . . . . . . . . . 14 (2o ∖ {∅}) = (({1o} ∪ 1o) ∖ 1o)
30 difun2 4425 . . . . . . . . . . . . . . 15 (({1o} ∪ 1o) ∖ 1o) = ({1o} ∖ 1o)
31 incom 4175 . . . . . . . . . . . . . . . . 17 ({1o} ∩ 1o) = (1o ∩ {1o})
32 1on 8098 . . . . . . . . . . . . . . . . . . 19 1o ∈ On
3332onordi 6288 . . . . . . . . . . . . . . . . . 18 Ord 1o
34 orddisj 6222 . . . . . . . . . . . . . . . . . 18 (Ord 1o → (1o ∩ {1o}) = ∅)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 (1o ∩ {1o}) = ∅
3631, 35eqtri 2841 . . . . . . . . . . . . . . . 16 ({1o} ∩ 1o) = ∅
37 disj3 4399 . . . . . . . . . . . . . . . 16 (({1o} ∩ 1o) = ∅ ↔ {1o} = ({1o} ∖ 1o))
3836, 37mpbi 231 . . . . . . . . . . . . . . 15 {1o} = ({1o} ∖ 1o)
3930, 38eqtr4i 2844 . . . . . . . . . . . . . 14 (({1o} ∪ 1o) ∖ 1o) = {1o}
4029, 39eqtri 2841 . . . . . . . . . . . . 13 (2o ∖ {∅}) = {1o}
4140imaeq2i 5920 . . . . . . . . . . . 12 (𝑦 “ (2o ∖ {∅})) = (𝑦 “ {1o})
4222, 41syl6eq 2869 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 supp ∅) = (𝑦 “ {1o}))
4342eleq1d 2894 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (𝑦 “ {1o}) ∈ Fin))
44 cnvimass 5942 . . . . . . . . . . . 12 (𝑦 “ {1o}) ⊆ dom 𝑦
4544, 20fssdm 6523 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 “ {1o}) ⊆ 𝐴)
4645biantrurd 533 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → ((𝑦 “ {1o}) ∈ Fin ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin)))
4717, 43, 463bitrd 306 . . . . . . . . 9 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 finSupp ∅ ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin)))
48 elfpw 8814 . . . . . . . . 9 ((𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin))
4947, 48syl6bbr 290 . . . . . . . 8 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)))
5049rabbidva 3476 . . . . . . 7 (𝐴𝑉 → {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2om 𝐴) ∣ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)})
51 cnveq 5737 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
5251imaeq1d 5921 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 “ {1o}) = (𝑦 “ {1o}))
5352cbvmptv 5160 . . . . . . . 8 (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) = (𝑦 ∈ (2om 𝐴) ↦ (𝑦 “ {1o}))
5453mptpreima 6085 . . . . . . 7 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2om 𝐴) ∣ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)}
5550, 5, 543eqtr4g 2878 . . . . . 6 (𝐴𝑉𝑆 = ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)))
5655imaeq2d 5922 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))))
57 f1ofo 6615 . . . . . . 7 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–1-1-onto→𝒫 𝐴 → (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–onto→𝒫 𝐴)
582, 57syl 17 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–onto→𝒫 𝐴)
59 inss1 4202 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
60 foimacnv 6625 . . . . . 6 (((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–onto→𝒫 𝐴 ∧ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴) → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
6158, 59, 60sylancl 586 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
6256, 61eqtrd 2853 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
63 f1oeq3 6599 . . . 4 (((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = (𝒫 𝐴 ∩ Fin) → (((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) ↔ ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→(𝒫 𝐴 ∩ Fin)))
6462, 63syl 17 . . 3 (𝐴𝑉 → (((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) ↔ ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→(𝒫 𝐴 ∩ Fin)))
65 resmpt 5898 . . . . . 6 (𝑆 ⊆ (2om 𝐴) → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆) = (𝑥𝑆 ↦ (𝑥 “ {1o})))
667, 65ax-mp 5 . . . . 5 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆) = (𝑥𝑆 ↦ (𝑥 “ {1o}))
67 pwfi2f1o.f . . . . 5 𝐹 = (𝑥𝑆 ↦ (𝑥 “ {1o}))
6866, 67eqtr4i 2844 . . . 4 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆) = 𝐹
69 f1oeq1 6597 . . . 4 (((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆) = 𝐹 → (((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→(𝒫 𝐴 ∩ Fin) ↔ 𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin)))
7068, 69mp1i 13 . . 3 (𝐴𝑉 → (((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→(𝒫 𝐴 ∩ Fin) ↔ 𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin)))
7164, 70bitrd 280 . 2 (𝐴𝑉 → (((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) ↔ 𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin)))
729, 71mpbid 233 1 (𝐴𝑉𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  cdif 3930  cun 3931  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  {csn 4557   class class class wbr 5057  cmpt 5137  ccnv 5547  cres 5550  cima 5551  Ord word 6183  suc csuc 6186  Fun wfun 6342  wf 6344  1-1wf1 6345  ontowfo 6346  1-1-ontowf1o 6347  (class class class)co 7145   supp csupp 7819  1oc1o 8084  2oc2o 8085  m cmap 8395  Fincfn 8497   finSupp cfsupp 8821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-supp 7820  df-1o 8091  df-2o 8092  df-map 8397  df-fsupp 8822
This theorem is referenced by:  pwfi2en  39575
  Copyright terms: Public domain W3C validator