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Theorem pwfi2f1o 41838
Description: The pw2f1o 9077 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 2733 . . . . 5 (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) = (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o}))
21pw2f1o2 41777 . . . 4 (𝐴𝑉 → (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–1-1-onto→𝒫 𝐴)
3 f1of1 6833 . . . 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 4078 . . . 4 {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2om 𝐴)
75, 6eqsstri 4017 . . 3 𝑆 ⊆ (2om 𝐴)
8 f1ores 6848 . . 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 587 . 2 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆))
10 elmapfun 8860 . . . . . . . . . . . . 13 (𝑦 ∈ (2om 𝐴) → Fun 𝑦)
11 id 22 . . . . . . . . . . . . 13 (𝑦 ∈ (2om 𝐴) → 𝑦 ∈ (2om 𝐴))
12 0ex 5308 . . . . . . . . . . . . . 14 ∅ ∈ V
1312a1i 11 . . . . . . . . . . . . 13 (𝑦 ∈ (2om 𝐴) → ∅ ∈ V)
1410, 11, 133jca 1129 . . . . . . . . . . . 12 (𝑦 ∈ (2om 𝐴) → (Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V))
1514adantl 483 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V))
16 funisfsupp 9367 . . . . . . . . . . 11 ((Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1715, 16syl 17 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1813anim2i 618 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝐴𝑉 ∧ ∅ ∈ V))
19 elmapi 8843 . . . . . . . . . . . . . 14 (𝑦 ∈ (2om 𝐴) → 𝑦:𝐴⟶2o)
2019adantl 483 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → 𝑦:𝐴⟶2o)
21 fsuppeq 8160 . . . . . . . . . . . . 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 8467 . . . . . . . . . . . . . . . 16 2o = suc 1o
24 df-suc 6371 . . . . . . . . . . . . . . . . 17 suc 1o = (1o ∪ {1o})
2524equncomi 4156 . . . . . . . . . . . . . . . 16 suc 1o = ({1o} ∪ 1o)
2623, 25eqtri 2761 . . . . . . . . . . . . . . 15 2o = ({1o} ∪ 1o)
27 df1o2 8473 . . . . . . . . . . . . . . . 16 1o = {∅}
2827eqcomi 2742 . . . . . . . . . . . . . . 15 {∅} = 1o
2926, 28difeq12i 4121 . . . . . . . . . . . . . 14 (2o ∖ {∅}) = (({1o} ∪ 1o) ∖ 1o)
30 difun2 4481 . . . . . . . . . . . . . . 15 (({1o} ∪ 1o) ∖ 1o) = ({1o} ∖ 1o)
31 incom 4202 . . . . . . . . . . . . . . . . 17 ({1o} ∩ 1o) = (1o ∩ {1o})
32 1on 8478 . . . . . . . . . . . . . . . . . . 19 1o ∈ On
3332onordi 6476 . . . . . . . . . . . . . . . . . 18 Ord 1o
34 orddisj 6403 . . . . . . . . . . . . . . . . . 18 (Ord 1o → (1o ∩ {1o}) = ∅)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 (1o ∩ {1o}) = ∅
3631, 35eqtri 2761 . . . . . . . . . . . . . . . 16 ({1o} ∩ 1o) = ∅
37 disj3 4454 . . . . . . . . . . . . . . . 16 (({1o} ∩ 1o) = ∅ ↔ {1o} = ({1o} ∖ 1o))
3836, 37mpbi 229 . . . . . . . . . . . . . . 15 {1o} = ({1o} ∖ 1o)
3930, 38eqtr4i 2764 . . . . . . . . . . . . . 14 (({1o} ∪ 1o) ∖ 1o) = {1o}
4029, 39eqtri 2761 . . . . . . . . . . . . 13 (2o ∖ {∅}) = {1o}
4140imaeq2i 6058 . . . . . . . . . . . 12 (𝑦 “ (2o ∖ {∅})) = (𝑦 “ {1o})
4222, 41eqtrdi 2789 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 supp ∅) = (𝑦 “ {1o}))
4342eleq1d 2819 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (𝑦 “ {1o}) ∈ Fin))
44 cnvimass 6081 . . . . . . . . . . . 12 (𝑦 “ {1o}) ⊆ dom 𝑦
4544, 20fssdm 6738 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 “ {1o}) ⊆ 𝐴)
4645biantrurd 534 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → ((𝑦 “ {1o}) ∈ Fin ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin)))
4717, 43, 463bitrd 305 . . . . . . . . 9 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 finSupp ∅ ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin)))
48 elfpw 9354 . . . . . . . . 9 ((𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin))
4947, 48bitr4di 289 . . . . . . . 8 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)))
5049rabbidva 3440 . . . . . . 7 (𝐴𝑉 → {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2om 𝐴) ∣ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)})
51 cnveq 5874 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
5251imaeq1d 6059 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 “ {1o}) = (𝑦 “ {1o}))
5352cbvmptv 5262 . . . . . . . 8 (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) = (𝑦 ∈ (2om 𝐴) ↦ (𝑦 “ {1o}))
5453mptpreima 6238 . . . . . . 7 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2om 𝐴) ∣ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)}
5550, 5, 543eqtr4g 2798 . . . . . 6 (𝐴𝑉𝑆 = ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)))
5655imaeq2d 6060 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))))
57 f1ofo 6841 . . . . . . 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 4229 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
60 foimacnv 6851 . . . . . 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 587 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
6256, 61eqtrd 2773 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
63 f1oeq3 6824 . . . 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 6038 . . . . . 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 2764 . . . 4 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆) = 𝐹
69 f1oeq1 6822 . . . 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 279 . 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 231 1 (𝐴𝑉𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  cdif 3946  cun 3947  cin 3948  wss 3949  c0 4323  𝒫 cpw 4603  {csn 4629   class class class wbr 5149  cmpt 5232  ccnv 5676  cres 5679  cima 5680  Ord word 6364  suc csuc 6367  Fun wfun 6538  wf 6540  1-1wf1 6541  ontowfo 6542  1-1-ontowf1o 6543  (class class class)co 7409   supp csupp 8146  1oc1o 8459  2oc2o 8460  m cmap 8820  Fincfn 8939   finSupp cfsupp 9361
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-supp 8147  df-1o 8466  df-2o 8467  df-map 8822  df-fsupp 9362
This theorem is referenced by:  pwfi2en  41839
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