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Theorem pwfi2f1o 41452
Description: The pw2f1o 9028 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 2737 . . . . 5 (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) = (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o}))
21pw2f1o2 41391 . . . 4 (𝐴𝑉 → (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–1-1-onto→𝒫 𝐴)
3 f1of1 6788 . . . 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 4042 . . . 4 {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2om 𝐴)
75, 6eqsstri 3983 . . 3 𝑆 ⊆ (2om 𝐴)
8 f1ores 6803 . . 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 8811 . . . . . . . . . . . . 13 (𝑦 ∈ (2om 𝐴) → Fun 𝑦)
11 id 22 . . . . . . . . . . . . 13 (𝑦 ∈ (2om 𝐴) → 𝑦 ∈ (2om 𝐴))
12 0ex 5269 . . . . . . . . . . . . . 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 9317 . . . . . . . . . . 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 8794 . . . . . . . . . . . . . 14 (𝑦 ∈ (2om 𝐴) → 𝑦:𝐴⟶2o)
2019adantl 483 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → 𝑦:𝐴⟶2o)
21 fsuppeq 8111 . . . . . . . . . . . . 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 8418 . . . . . . . . . . . . . . . 16 2o = suc 1o
24 df-suc 6328 . . . . . . . . . . . . . . . . 17 suc 1o = (1o ∪ {1o})
2524equncomi 4120 . . . . . . . . . . . . . . . 16 suc 1o = ({1o} ∪ 1o)
2623, 25eqtri 2765 . . . . . . . . . . . . . . 15 2o = ({1o} ∪ 1o)
27 df1o2 8424 . . . . . . . . . . . . . . . 16 1o = {∅}
2827eqcomi 2746 . . . . . . . . . . . . . . 15 {∅} = 1o
2926, 28difeq12i 4085 . . . . . . . . . . . . . 14 (2o ∖ {∅}) = (({1o} ∪ 1o) ∖ 1o)
30 difun2 4445 . . . . . . . . . . . . . . 15 (({1o} ∪ 1o) ∖ 1o) = ({1o} ∖ 1o)
31 incom 4166 . . . . . . . . . . . . . . . . 17 ({1o} ∩ 1o) = (1o ∩ {1o})
32 1on 8429 . . . . . . . . . . . . . . . . . . 19 1o ∈ On
3332onordi 6433 . . . . . . . . . . . . . . . . . 18 Ord 1o
34 orddisj 6360 . . . . . . . . . . . . . . . . . 18 (Ord 1o → (1o ∩ {1o}) = ∅)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 (1o ∩ {1o}) = ∅
3631, 35eqtri 2765 . . . . . . . . . . . . . . . 16 ({1o} ∩ 1o) = ∅
37 disj3 4418 . . . . . . . . . . . . . . . 16 (({1o} ∩ 1o) = ∅ ↔ {1o} = ({1o} ∖ 1o))
3836, 37mpbi 229 . . . . . . . . . . . . . . 15 {1o} = ({1o} ∖ 1o)
3930, 38eqtr4i 2768 . . . . . . . . . . . . . 14 (({1o} ∪ 1o) ∖ 1o) = {1o}
4029, 39eqtri 2765 . . . . . . . . . . . . 13 (2o ∖ {∅}) = {1o}
4140imaeq2i 6016 . . . . . . . . . . . 12 (𝑦 “ (2o ∖ {∅})) = (𝑦 “ {1o})
4222, 41eqtrdi 2793 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 supp ∅) = (𝑦 “ {1o}))
4342eleq1d 2823 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (𝑦 “ {1o}) ∈ Fin))
44 cnvimass 6038 . . . . . . . . . . . 12 (𝑦 “ {1o}) ⊆ dom 𝑦
4544, 20fssdm 6693 . . . . . . . . . . 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 9305 . . . . . . . . 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 3417 . . . . . . 7 (𝐴𝑉 → {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2om 𝐴) ∣ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)})
51 cnveq 5834 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
5251imaeq1d 6017 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 “ {1o}) = (𝑦 “ {1o}))
5352cbvmptv 5223 . . . . . . . 8 (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) = (𝑦 ∈ (2om 𝐴) ↦ (𝑦 “ {1o}))
5453mptpreima 6195 . . . . . . 7 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2om 𝐴) ∣ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)}
5550, 5, 543eqtr4g 2802 . . . . . 6 (𝐴𝑉𝑆 = ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)))
5655imaeq2d 6018 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))))
57 f1ofo 6796 . . . . . . 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 4193 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
60 foimacnv 6806 . . . . . 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 2777 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
63 f1oeq3 6779 . . . 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 5996 . . . . . 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 2768 . . . 4 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆) = 𝐹
69 f1oeq1 6777 . . . 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 3410  Vcvv 3448  cdif 3912  cun 3913  cin 3914  wss 3915  c0 4287  𝒫 cpw 4565  {csn 4591   class class class wbr 5110  cmpt 5193  ccnv 5637  cres 5640  cima 5641  Ord word 6321  suc csuc 6324  Fun wfun 6495  wf 6497  1-1wf1 6498  ontowfo 6499  1-1-ontowf1o 6500  (class class class)co 7362   supp csupp 8097  1oc1o 8410  2oc2o 8411  m cmap 8772  Fincfn 8890   finSupp cfsupp 9312
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-supp 8098  df-1o 8417  df-2o 8418  df-map 8774  df-fsupp 9313
This theorem is referenced by:  pwfi2en  41453
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