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Theorem pwfi2f1o 43085
Description: The pw2f1o 9116 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 2735 . . . . 5 (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) = (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o}))
21pw2f1o2 43027 . . . 4 (𝐴𝑉 → (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})):(2om 𝐴)–1-1-onto→𝒫 𝐴)
3 f1of1 6848 . . . 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 4090 . . . 4 {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2om 𝐴)
75, 6eqsstri 4030 . . 3 𝑆 ⊆ (2om 𝐴)
8 f1ores 6863 . . 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 8905 . . . . . . . . . . . . 13 (𝑦 ∈ (2om 𝐴) → Fun 𝑦)
11 id 22 . . . . . . . . . . . . 13 (𝑦 ∈ (2om 𝐴) → 𝑦 ∈ (2om 𝐴))
12 0ex 5313 . . . . . . . . . . . . . 14 ∅ ∈ V
1312a1i 11 . . . . . . . . . . . . 13 (𝑦 ∈ (2om 𝐴) → ∅ ∈ V)
1410, 11, 133jca 1127 . . . . . . . . . . . 12 (𝑦 ∈ (2om 𝐴) → (Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V))
1514adantl 481 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V))
16 funisfsupp 9405 . . . . . . . . . . 11 ((Fun 𝑦𝑦 ∈ (2om 𝐴) ∧ ∅ ∈ V) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1715, 16syl 17 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1813anim2i 617 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝐴𝑉 ∧ ∅ ∈ V))
19 elmapi 8888 . . . . . . . . . . . . . 14 (𝑦 ∈ (2om 𝐴) → 𝑦:𝐴⟶2o)
2019adantl 481 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → 𝑦:𝐴⟶2o)
21 fsuppeq 8199 . . . . . . . . . . . . 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 8506 . . . . . . . . . . . . . . . 16 2o = suc 1o
24 df-suc 6392 . . . . . . . . . . . . . . . . 17 suc 1o = (1o ∪ {1o})
2524equncomi 4170 . . . . . . . . . . . . . . . 16 suc 1o = ({1o} ∪ 1o)
2623, 25eqtri 2763 . . . . . . . . . . . . . . 15 2o = ({1o} ∪ 1o)
27 df1o2 8512 . . . . . . . . . . . . . . . 16 1o = {∅}
2827eqcomi 2744 . . . . . . . . . . . . . . 15 {∅} = 1o
2926, 28difeq12i 4134 . . . . . . . . . . . . . 14 (2o ∖ {∅}) = (({1o} ∪ 1o) ∖ 1o)
30 difun2 4487 . . . . . . . . . . . . . . 15 (({1o} ∪ 1o) ∖ 1o) = ({1o} ∖ 1o)
31 incom 4217 . . . . . . . . . . . . . . . . 17 ({1o} ∩ 1o) = (1o ∩ {1o})
32 1on 8517 . . . . . . . . . . . . . . . . . . 19 1o ∈ On
3332onordi 6497 . . . . . . . . . . . . . . . . . 18 Ord 1o
34 orddisj 6424 . . . . . . . . . . . . . . . . . 18 (Ord 1o → (1o ∩ {1o}) = ∅)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 (1o ∩ {1o}) = ∅
3631, 35eqtri 2763 . . . . . . . . . . . . . . . 16 ({1o} ∩ 1o) = ∅
37 disj3 4460 . . . . . . . . . . . . . . . 16 (({1o} ∩ 1o) = ∅ ↔ {1o} = ({1o} ∖ 1o))
3836, 37mpbi 230 . . . . . . . . . . . . . . 15 {1o} = ({1o} ∖ 1o)
3930, 38eqtr4i 2766 . . . . . . . . . . . . . 14 (({1o} ∪ 1o) ∖ 1o) = {1o}
4029, 39eqtri 2763 . . . . . . . . . . . . 13 (2o ∖ {∅}) = {1o}
4140imaeq2i 6078 . . . . . . . . . . . 12 (𝑦 “ (2o ∖ {∅})) = (𝑦 “ {1o})
4222, 41eqtrdi 2791 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 supp ∅) = (𝑦 “ {1o}))
4342eleq1d 2824 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (𝑦 “ {1o}) ∈ Fin))
44 cnvimass 6102 . . . . . . . . . . . 12 (𝑦 “ {1o}) ⊆ dom 𝑦
4544, 20fssdm 6756 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2om 𝐴)) → (𝑦 “ {1o}) ⊆ 𝐴)
4645biantrurd 532 . . . . . . . . . 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 9392 . . . . . . . . 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 5887 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
5251imaeq1d 6079 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 “ {1o}) = (𝑦 “ {1o}))
5352cbvmptv 5261 . . . . . . . 8 (𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) = (𝑦 ∈ (2om 𝐴) ↦ (𝑦 “ {1o}))
5453mptpreima 6260 . . . . . . 7 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2om 𝐴) ∣ (𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)}
5550, 5, 543eqtr4g 2800 . . . . . 6 (𝐴𝑉𝑆 = ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)))
5655imaeq2d 6080 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))))
57 f1ofo 6856 . . . . . . 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 4245 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
60 foimacnv 6866 . . . . . 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 2775 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
63 f1oeq3 6839 . . . 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 6057 . . . . . 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 2766 . . . 4 ((𝑥 ∈ (2om 𝐴) ↦ (𝑥 “ {1o})) ↾ 𝑆) = 𝐹
69 f1oeq1 6837 . . . 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 232 1 (𝐴𝑉𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  cmpt 5231  ccnv 5688  cres 5691  cima 5692  Ord word 6385  suc csuc 6388  Fun wfun 6557  wf 6559  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  (class class class)co 7431   supp csupp 8184  1oc1o 8498  2oc2o 8499  m cmap 8865  Fincfn 8984   finSupp cfsupp 9399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-supp 8185  df-1o 8505  df-2o 8506  df-map 8867  df-fsupp 9400
This theorem is referenced by:  pwfi2en  43086
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