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Theorem pwfi2f1o 38367
Description: The pw2f1o 8276 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 2765 . . . . 5 (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))
21pw2f1o2 38306 . . . 4 (𝐴𝑉 → (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})):(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴)
3 f1of1 6323 . . . 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 3849 . . . 4 {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2𝑜𝑚 𝐴)
75, 6eqsstri 3797 . . 3 𝑆 ⊆ (2𝑜𝑚 𝐴)
8 f1ores 6338 . . 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 580 . 2 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆))
10 elmapfun 8088 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → Fun 𝑦)
11 id 22 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → 𝑦 ∈ (2𝑜𝑚 𝐴))
12 0ex 4952 . . . . . . . . . . . . . 14 ∅ ∈ V
1312a1i 11 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → ∅ ∈ V)
1410, 11, 133jca 1158 . . . . . . . . . . . 12 (𝑦 ∈ (2𝑜𝑚 𝐴) → (Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V))
1514adantl 473 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V))
16 funisfsupp 8491 . . . . . . . . . . 11 ((Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1715, 16syl 17 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1813anim2i 610 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝐴𝑉 ∧ ∅ ∈ V))
19 elmapi 8086 . . . . . . . . . . . . . 14 (𝑦 ∈ (2𝑜𝑚 𝐴) → 𝑦:𝐴⟶2𝑜)
2019adantl 473 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → 𝑦:𝐴⟶2𝑜)
21 frnsuppeq 7513 . . . . . . . . . . . . 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 7769 . . . . . . . . . . . . . . . 16 2𝑜 = suc 1𝑜
24 df-suc 5916 . . . . . . . . . . . . . . . . 17 suc 1𝑜 = (1𝑜 ∪ {1𝑜})
2524equncomi 3923 . . . . . . . . . . . . . . . 16 suc 1𝑜 = ({1𝑜} ∪ 1𝑜)
2623, 25eqtri 2787 . . . . . . . . . . . . . . 15 2𝑜 = ({1𝑜} ∪ 1𝑜)
27 df1o2 7781 . . . . . . . . . . . . . . . 16 1𝑜 = {∅}
2827eqcomi 2774 . . . . . . . . . . . . . . 15 {∅} = 1𝑜
2926, 28difeq12i 3890 . . . . . . . . . . . . . 14 (2𝑜 ∖ {∅}) = (({1𝑜} ∪ 1𝑜) ∖ 1𝑜)
30 difun2 4210 . . . . . . . . . . . . . . 15 (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = ({1𝑜} ∖ 1𝑜)
31 incom 3969 . . . . . . . . . . . . . . . . 17 ({1𝑜} ∩ 1𝑜) = (1𝑜 ∩ {1𝑜})
32 1on 7775 . . . . . . . . . . . . . . . . . . 19 1𝑜 ∈ On
3332onordi 6014 . . . . . . . . . . . . . . . . . 18 Ord 1𝑜
34 orddisj 5948 . . . . . . . . . . . . . . . . . 18 (Ord 1𝑜 → (1𝑜 ∩ {1𝑜}) = ∅)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 (1𝑜 ∩ {1𝑜}) = ∅
3631, 35eqtri 2787 . . . . . . . . . . . . . . . 16 ({1𝑜} ∩ 1𝑜) = ∅
37 disj3 4184 . . . . . . . . . . . . . . . 16 (({1𝑜} ∩ 1𝑜) = ∅ ↔ {1𝑜} = ({1𝑜} ∖ 1𝑜))
3836, 37mpbi 221 . . . . . . . . . . . . . . 15 {1𝑜} = ({1𝑜} ∖ 1𝑜)
3930, 38eqtr4i 2790 . . . . . . . . . . . . . 14 (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = {1𝑜}
4029, 39eqtri 2787 . . . . . . . . . . . . 13 (2𝑜 ∖ {∅}) = {1𝑜}
4140imaeq2i 5648 . . . . . . . . . . . 12 (𝑦 “ (2𝑜 ∖ {∅})) = (𝑦 “ {1𝑜})
4222, 41syl6eq 2815 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 supp ∅) = (𝑦 “ {1𝑜}))
4342eleq1d 2829 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (𝑦 “ {1𝑜}) ∈ Fin))
44 cnvimass 5669 . . . . . . . . . . . 12 (𝑦 “ {1𝑜}) ⊆ dom 𝑦
4544, 20fssdm 6241 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 “ {1𝑜}) ⊆ 𝐴)
4645biantrurd 528 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → ((𝑦 “ {1𝑜}) ∈ Fin ↔ ((𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (𝑦 “ {1𝑜}) ∈ Fin)))
4717, 43, 463bitrd 296 . . . . . . . . 9 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ ((𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (𝑦 “ {1𝑜}) ∈ Fin)))
48 elfpw 8479 . . . . . . . . 9 ((𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (𝑦 “ {1𝑜}) ∈ Fin))
4947, 48syl6bbr 280 . . . . . . . 8 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)))
5049rabbidva 3337 . . . . . . 7 (𝐴𝑉 → {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)})
51 cnveq 5466 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
5251imaeq1d 5649 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 “ {1𝑜}) = (𝑦 “ {1𝑜}))
5352cbvmptv 4911 . . . . . . . 8 (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) = (𝑦 ∈ (2𝑜𝑚 𝐴) ↦ (𝑦 “ {1𝑜}))
5453mptpreima 5816 . . . . . . 7 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)}
5550, 5, 543eqtr4g 2824 . . . . . 6 (𝐴𝑉𝑆 = ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)))
5655imaeq2d 5650 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆) = ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))))
57 f1ofo 6331 . . . . . . 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 3994 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
60 foimacnv 6341 . . . . . 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 580 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
6256, 61eqtrd 2799 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
63 f1oeq3 6316 . . . 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 5628 . . . . . 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 2790 . . . 4 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆) = 𝐹
69 f1oeq1 6314 . . . 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 270 . 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 223 1 (𝐴𝑉𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  cdif 3731  cun 3732  cin 3733  wss 3734  c0 4081  𝒫 cpw 4317  {csn 4336   class class class wbr 4811  cmpt 4890  ccnv 5278  cres 5281  cima 5282  Ord word 5909  suc csuc 5912  Fun wfun 6064  wf 6066  1-1wf1 6067  ontowfo 6068  1-1-ontowf1o 6069  (class class class)co 6846   supp csupp 7501  1𝑜c1o 7761  2𝑜c2o 7762  𝑚 cmap 8064  Fincfn 8164   finSupp cfsupp 8486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-ord 5913  df-on 5914  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-1st 7370  df-2nd 7371  df-supp 7502  df-1o 7768  df-2o 7769  df-map 8066  df-fsupp 8487
This theorem is referenced by:  pwfi2en  38368
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