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Theorem pwfi2f1o 42664
Description: The pw2f1o 9105 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 2725 . . . . 5 (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) = (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o}))
21pw2f1o2 42603 . . . 4 (𝐴𝑉 → (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–1-1-onto→𝒫 𝐴)
3 f1of1 6837 . . . 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 4073 . . . 4 {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2om 𝐴)
75, 6eqsstri 4011 . . 3 𝑆 ⊆ (2om 𝐴)
8 f1ores 6852 . . 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 584 . 2 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆))
10 elmapfun 8885 . . . . . . . . . . . . 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 1125 . . . . . . . . . . . 12 (𝑦 ∈ (2om 𝐴) → (Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V))
1514adantl 480 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V))
16 funisfsupp 9398 . . . . . . . . . . 11 ((Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1715, 16syl 17 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1813anim2i 615 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝐴𝑉 ∧ ∅ ∈ V))
19 elmapi 8868 . . . . . . . . . . . . . 14 (𝑦 ∈ (2om 𝐴) → 𝑦:𝐴⟶2o)
2019adantl 480 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → 𝑦:𝐴⟶2o)
21 fsuppeq 8180 . . . . . . . . . . . . 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 8488 . . . . . . . . . . . . . . . 16 2o = suc 1o
24 df-suc 6377 . . . . . . . . . . . . . . . . 17 suc 1o = (1o ∪ {1o})
2524equncomi 4152 . . . . . . . . . . . . . . . 16 suc 1o = ({1o} ∪ 1o)
2623, 25eqtri 2753 . . . . . . . . . . . . . . 15 2o = ({1o} ∪ 1o)
27 df1o2 8494 . . . . . . . . . . . . . . . 16 1o = {∅}
2827eqcomi 2734 . . . . . . . . . . . . . . 15 {∅} = 1o
2926, 28difeq12i 4116 . . . . . . . . . . . . . 14 (2o ∖ {∅}) = (({1o} ∪ 1o) ∖ 1o)
30 difun2 4482 . . . . . . . . . . . . . . 15 (({1o} ∪ 1o) ∖ 1o) = ({1o} ∖ 1o)
31 incom 4199 . . . . . . . . . . . . . . . . 17 ({1o} ∩ 1o) = (1o ∩ {1o})
32 1on 8499 . . . . . . . . . . . . . . . . . . 19 1o ∈ On
3332onordi 6482 . . . . . . . . . . . . . . . . . 18 Ord 1o
34 orddisj 6409 . . . . . . . . . . . . . . . . . 18 (Ord 1o → (1o ∩ {1o}) = ∅)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 (1o ∩ {1o}) = ∅
3631, 35eqtri 2753 . . . . . . . . . . . . . . . 16 ({1o} ∩ 1o) = ∅
37 disj3 4455 . . . . . . . . . . . . . . . 16 (({1o} ∩ 1o) = ∅ ↔ {1o} = ({1o} ∖ 1o))
3836, 37mpbi 229 . . . . . . . . . . . . . . 15 {1o} = ({1o} ∖ 1o)
3930, 38eqtr4i 2756 . . . . . . . . . . . . . 14 (({1o} ∪ 1o) ∖ 1o) = {1o}
4029, 39eqtri 2753 . . . . . . . . . . . . 13 (2o ∖ {∅}) = {1o}
4140imaeq2i 6062 . . . . . . . . . . . 12 (𝑦 “ (2o ∖ {∅})) = (𝑦 “ {1o})
4222, 41eqtrdi 2781 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 supp ∅) = (𝑦 “ {1o}))
4342eleq1d 2810 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (𝑦 “ {1o}) ∈ Fin))
44 cnvimass 6086 . . . . . . . . . . . 12 (𝑦 “ {1o}) ⊆ dom 𝑦
4544, 20fssdm 6742 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 “ {1o}) ⊆ 𝐴)
4645biantrurd 531 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → ((𝑦 “ {1o}) ∈ Fin ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin)))
4717, 43, 463bitrd 304 . . . . . . . . 9 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 finSupp ∅ ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin)))
48 elfpw 9385 . . . . . . . . 9 ((𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑦 “ {1o}) ⊆ 𝐴 ∧ (𝑦 “ {1o}) ∈ Fin))
4947, 48bitr4di 288 . . . . . . . 8 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)))
5049rabbidva 3425 . . . . . . 7 (𝐴𝑉 → {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2om 𝐴) ∣ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)})
51 cnveq 5876 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
5251imaeq1d 6063 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 “ {1o}) = (𝑦 “ {1o}))
5352cbvmptv 5262 . . . . . . . 8 (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) = (𝑦 ∈ (2om 𝐴) ↦ (𝑦 “ {1o}))
5453mptpreima 6244 . . . . . . 7 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2om 𝐴) ∣ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)}
5550, 5, 543eqtr4g 2790 . . . . . 6 (𝐴𝑉𝑆 = ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)))
5655imaeq2d 6064 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))))
57 f1ofo 6845 . . . . . . 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 4227 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
60 foimacnv 6855 . . . . . 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 584 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
6256, 61eqtrd 2765 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
63 f1oeq3 6828 . . . 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 6042 . . . . . 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 2756 . . . 4 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆) = 𝐹
69 f1oeq1 6826 . . . 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 278 . 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 394  w3a 1084   = wceq 1533  wcel 2098  {crab 3418  Vcvv 3461  cdif 3941  cun 3942  cin 3943  wss 3944  c0 4322  𝒫 cpw 4604  {csn 4630   class class class wbr 5149  cmpt 5232  ccnv 5677  cres 5680  cima 5681  Ord word 6370  suc csuc 6373  Fun wfun 6543  wf 6545  1-1wf1 6546  ontowfo 6547  1-1-ontowf1o 6548  (class class class)co 7419   supp csupp 8165  1oc1o 8480  2oc2o 8481  m cmap 8845  Fincfn 8964   finSupp cfsupp 9392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-supp 8166  df-1o 8487  df-2o 8488  df-map 8847  df-fsupp 9393
This theorem is referenced by:  pwfi2en  42665
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