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 39126
Description: The pw2f1o 8417 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 2773 . . . . 5 (𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) = (𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o}))
21pw2f1o2 39065 . . . 4 (𝐴𝑉 → (𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})):(2o𝑚 𝐴)–1-1-onto→𝒫 𝐴)
3 f1of1 6441 . . . 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 3941 . . . 4 {𝑦 ∈ (2o𝑚 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2o𝑚 𝐴)
75, 6eqsstri 3886 . . 3 𝑆 ⊆ (2o𝑚 𝐴)
8 f1ores 6456 . . 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 578 . 2 (𝐴𝑉 → ((𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆))
10 elmapfun 8229 . . . . . . . . . . . . 13 (𝑦 ∈ (2o𝑚 𝐴) → Fun 𝑦)
11 id 22 . . . . . . . . . . . . 13 (𝑦 ∈ (2o𝑚 𝐴) → 𝑦 ∈ (2o𝑚 𝐴))
12 0ex 5065 . . . . . . . . . . . . . 14 ∅ ∈ V
1312a1i 11 . . . . . . . . . . . . 13 (𝑦 ∈ (2o𝑚 𝐴) → ∅ ∈ V)
1410, 11, 133jca 1109 . . . . . . . . . . . 12 (𝑦 ∈ (2o𝑚 𝐴) → (Fun 𝑦𝑦 ∈ (2o𝑚 𝐴) ∧ ∅ ∈ V))
1514adantl 474 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2o𝑚 𝐴)) → (Fun 𝑦𝑦 ∈ (2o𝑚 𝐴) ∧ ∅ ∈ V))
16 funisfsupp 8632 . . . . . . . . . . 11 ((Fun 𝑦𝑦 ∈ (2o𝑚 𝐴) ∧ ∅ ∈ V) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1715, 16syl 17 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2o𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1813anim2i 608 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2o𝑚 𝐴)) → (𝐴𝑉 ∧ ∅ ∈ V))
19 elmapi 8227 . . . . . . . . . . . . . 14 (𝑦 ∈ (2o𝑚 𝐴) → 𝑦:𝐴⟶2o)
2019adantl 474 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2o𝑚 𝐴)) → 𝑦:𝐴⟶2o)
21 frnsuppeq 7644 . . . . . . . . . . . . 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 7905 . . . . . . . . . . . . . . . 16 2o = suc 1o
24 df-suc 6033 . . . . . . . . . . . . . . . . 17 suc 1o = (1o ∪ {1o})
2524equncomi 4015 . . . . . . . . . . . . . . . 16 suc 1o = ({1o} ∪ 1o)
2623, 25eqtri 2797 . . . . . . . . . . . . . . 15 2o = ({1o} ∪ 1o)
27 df1o2 7917 . . . . . . . . . . . . . . . 16 1o = {∅}
2827eqcomi 2782 . . . . . . . . . . . . . . 15 {∅} = 1o
2926, 28difeq12i 3982 . . . . . . . . . . . . . 14 (2o ∖ {∅}) = (({1o} ∪ 1o) ∖ 1o)
30 difun2 4307 . . . . . . . . . . . . . . 15 (({1o} ∪ 1o) ∖ 1o) = ({1o} ∖ 1o)
31 incom 4061 . . . . . . . . . . . . . . . . 17 ({1o} ∩ 1o) = (1o ∩ {1o})
32 1on 7911 . . . . . . . . . . . . . . . . . . 19 1o ∈ On
3332onordi 6131 . . . . . . . . . . . . . . . . . 18 Ord 1o
34 orddisj 6065 . . . . . . . . . . . . . . . . . 18 (Ord 1o → (1o ∩ {1o}) = ∅)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 (1o ∩ {1o}) = ∅
3631, 35eqtri 2797 . . . . . . . . . . . . . . . 16 ({1o} ∩ 1o) = ∅
37 disj3 4281 . . . . . . . . . . . . . . . 16 (({1o} ∩ 1o) = ∅ ↔ {1o} = ({1o} ∖ 1o))
3836, 37mpbi 222 . . . . . . . . . . . . . . 15 {1o} = ({1o} ∖ 1o)
3930, 38eqtr4i 2800 . . . . . . . . . . . . . 14 (({1o} ∪ 1o) ∖ 1o) = {1o}
4029, 39eqtri 2797 . . . . . . . . . . . . 13 (2o ∖ {∅}) = {1o}
4140imaeq2i 5766 . . . . . . . . . . . 12 (𝑦 “ (2o ∖ {∅})) = (𝑦 “ {1o})
4222, 41syl6eq 2825 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2o𝑚 𝐴)) → (𝑦 supp ∅) = (𝑦 “ {1o}))
4342eleq1d 2845 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2o𝑚 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (𝑦 “ {1o}) ∈ Fin))
44 cnvimass 5787 . . . . . . . . . . . 12 (𝑦 “ {1o}) ⊆ dom 𝑦
4544, 20fssdm 6358 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2o𝑚 𝐴)) → (𝑦 “ {1o}) ⊆ 𝐴)
4645biantrurd 525 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2o𝑚 𝐴)) → ((𝑦 “ {1o}) ∈ Fin ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin)))
4717, 43, 463bitrd 297 . . . . . . . . 9 ((𝐴𝑉𝑦 ∈ (2o𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin)))
48 elfpw 8620 . . . . . . . . 9 ((𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin))
4947, 48syl6bbr 281 . . . . . . . 8 ((𝐴𝑉𝑦 ∈ (2o𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)))
5049rabbidva 3397 . . . . . . 7 (𝐴𝑉 → {𝑦 ∈ (2o𝑚 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2o𝑚 𝐴) ∣ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)})
51 cnveq 5591 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
5251imaeq1d 5767 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 “ {1o}) = (𝑦 “ {1o}))
5352cbvmptv 5025 . . . . . . . 8 (𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) = (𝑦 ∈ (2o𝑚 𝐴) ↦ (𝑦 “ {1o}))
5453mptpreima 5929 . . . . . . 7 ((𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2o𝑚 𝐴) ∣ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)}
5550, 5, 543eqtr4g 2834 . . . . . 6 (𝐴𝑉𝑆 = ((𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)))
5655imaeq2d 5768 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = ((𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) “ ((𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))))
57 f1ofo 6449 . . . . . . 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 4087 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
60 foimacnv 6459 . . . . . 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 578 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) “ ((𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
6256, 61eqtrd 2809 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
63 f1oeq3 6433 . . . 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 5748 . . . . . 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 2800 . . . 4 ((𝑥 ∈ (2o𝑚 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆) = 𝐹
69 f1oeq1 6431 . . . 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 271 . 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 224 1 (𝐴𝑉𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069   = wceq 1508  wcel 2051  {crab 3087  Vcvv 3410  cdif 3821  cun 3822  cin 3823  wss 3824  c0 4173  𝒫 cpw 4417  {csn 4436   class class class wbr 4926  cmpt 5005  ccnv 5403  cres 5406  cima 5407  Ord word 6026  suc csuc 6029  Fun wfun 6180  wf 6182  1-1wf1 6183  ontowfo 6184  1-1-ontowf1o 6185  (class class class)co 6975   supp csupp 7632  1oc1o 7897  2oc2o 7898  𝑚 cmap 8205  Fincfn 8305   finSupp cfsupp 8627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-ord 6030  df-on 6031  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-ov 6978  df-oprab 6979  df-mpo 6980  df-1st 7500  df-2nd 7501  df-supp 7633  df-1o 7904  df-2o 7905  df-map 8207  df-fsupp 8628
This theorem is referenced by:  pwfi2en  39127
  Copyright terms: Public domain W3C validator