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Theorem pwfi2f1o 37983
Description: The pw2f1o 8106 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 2651 . . . . 5 (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))
21pw2f1o2 37922 . . . 4 (𝐴𝑉 → (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})):(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴)
3 f1of1 6174 . . . 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 3720 . . . 4 {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2𝑜𝑚 𝐴)
75, 6eqsstri 3668 . . 3 𝑆 ⊆ (2𝑜𝑚 𝐴)
8 f1ores 6189 . . 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 695 . 2 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆))
10 elmapfun 7923 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → Fun 𝑦)
11 id 22 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → 𝑦 ∈ (2𝑜𝑚 𝐴))
12 0ex 4823 . . . . . . . . . . . . . 14 ∅ ∈ V
1312a1i 11 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → ∅ ∈ V)
1410, 11, 133jca 1261 . . . . . . . . . . . 12 (𝑦 ∈ (2𝑜𝑚 𝐴) → (Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V))
1514adantl 481 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V))
16 funisfsupp 8321 . . . . . . . . . . 11 ((Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1715, 16syl 17 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1813anim2i 592 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝐴𝑉 ∧ ∅ ∈ V))
19 elmapi 7921 . . . . . . . . . . . . . 14 (𝑦 ∈ (2𝑜𝑚 𝐴) → 𝑦:𝐴⟶2𝑜)
2019adantl 481 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → 𝑦:𝐴⟶2𝑜)
21 frnsuppeq 7352 . . . . . . . . . . . . 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 7606 . . . . . . . . . . . . . . . 16 2𝑜 = suc 1𝑜
24 df-suc 5767 . . . . . . . . . . . . . . . . 17 suc 1𝑜 = (1𝑜 ∪ {1𝑜})
2524equncomi 3792 . . . . . . . . . . . . . . . 16 suc 1𝑜 = ({1𝑜} ∪ 1𝑜)
2623, 25eqtri 2673 . . . . . . . . . . . . . . 15 2𝑜 = ({1𝑜} ∪ 1𝑜)
27 df1o2 7617 . . . . . . . . . . . . . . . 16 1𝑜 = {∅}
2827eqcomi 2660 . . . . . . . . . . . . . . 15 {∅} = 1𝑜
2926, 28difeq12i 3759 . . . . . . . . . . . . . 14 (2𝑜 ∖ {∅}) = (({1𝑜} ∪ 1𝑜) ∖ 1𝑜)
30 difun2 4081 . . . . . . . . . . . . . . 15 (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = ({1𝑜} ∖ 1𝑜)
31 incom 3838 . . . . . . . . . . . . . . . . 17 ({1𝑜} ∩ 1𝑜) = (1𝑜 ∩ {1𝑜})
32 1on 7612 . . . . . . . . . . . . . . . . . . 19 1𝑜 ∈ On
3332onordi 5870 . . . . . . . . . . . . . . . . . 18 Ord 1𝑜
34 orddisj 5800 . . . . . . . . . . . . . . . . . 18 (Ord 1𝑜 → (1𝑜 ∩ {1𝑜}) = ∅)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 (1𝑜 ∩ {1𝑜}) = ∅
3631, 35eqtri 2673 . . . . . . . . . . . . . . . 16 ({1𝑜} ∩ 1𝑜) = ∅
37 disj3 4054 . . . . . . . . . . . . . . . 16 (({1𝑜} ∩ 1𝑜) = ∅ ↔ {1𝑜} = ({1𝑜} ∖ 1𝑜))
3836, 37mpbi 220 . . . . . . . . . . . . . . 15 {1𝑜} = ({1𝑜} ∖ 1𝑜)
3930, 38eqtr4i 2676 . . . . . . . . . . . . . 14 (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = {1𝑜}
4029, 39eqtri 2673 . . . . . . . . . . . . 13 (2𝑜 ∖ {∅}) = {1𝑜}
4140imaeq2i 5499 . . . . . . . . . . . 12 (𝑦 “ (2𝑜 ∖ {∅})) = (𝑦 “ {1𝑜})
4222, 41syl6eq 2701 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 supp ∅) = (𝑦 “ {1𝑜}))
4342eleq1d 2715 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (𝑦 “ {1𝑜}) ∈ Fin))
44 cnvimass 5520 . . . . . . . . . . . 12 (𝑦 “ {1𝑜}) ⊆ dom 𝑦
45 fdm 6089 . . . . . . . . . . . . 13 (𝑦:𝐴⟶2𝑜 → dom 𝑦 = 𝐴)
4620, 45syl 17 . . . . . . . . . . . 12 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → dom 𝑦 = 𝐴)
4744, 46syl5sseq 3686 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 “ {1𝑜}) ⊆ 𝐴)
4847biantrurd 528 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → ((𝑦 “ {1𝑜}) ∈ Fin ↔ ((𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (𝑦 “ {1𝑜}) ∈ Fin)))
4917, 43, 483bitrd 294 . . . . . . . . 9 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ ((𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (𝑦 “ {1𝑜}) ∈ Fin)))
50 elfpw 8309 . . . . . . . . 9 ((𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (𝑦 “ {1𝑜}) ∈ Fin))
5149, 50syl6bbr 278 . . . . . . . 8 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)))
5251rabbidva 3219 . . . . . . 7 (𝐴𝑉 → {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)})
53 cnveq 5328 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
5453imaeq1d 5500 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 “ {1𝑜}) = (𝑦 “ {1𝑜}))
5554cbvmptv 4783 . . . . . . . 8 (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) = (𝑦 ∈ (2𝑜𝑚 𝐴) ↦ (𝑦 “ {1𝑜}))
5655mptpreima 5666 . . . . . . 7 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)}
5752, 5, 563eqtr4g 2710 . . . . . 6 (𝐴𝑉𝑆 = ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)))
5857imaeq2d 5501 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆) = ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))))
59 f1ofo 6182 . . . . . . 7 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})):(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴 → (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})):(2𝑜𝑚 𝐴)–onto→𝒫 𝐴)
602, 59syl 17 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})):(2𝑜𝑚 𝐴)–onto→𝒫 𝐴)
61 inss1 3866 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
62 foimacnv 6192 . . . . . 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))
6360, 61, 62sylancl 695 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
6458, 63eqtrd 2685 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
65 f1oeq3 6167 . . . 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)))
6664, 65syl 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)))
67 resmpt 5484 . . . . . 6 (𝑆 ⊆ (2𝑜𝑚 𝐴) → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆) = (𝑥𝑆 ↦ (𝑥 “ {1𝑜})))
687, 67ax-mp 5 . . . . 5 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆) = (𝑥𝑆 ↦ (𝑥 “ {1𝑜}))
69 pwfi2f1o.f . . . . 5 𝐹 = (𝑥𝑆 ↦ (𝑥 “ {1𝑜}))
7068, 69eqtr4i 2676 . . . 4 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆) = 𝐹
71 f1oeq1 6165 . . . 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)))
7270, 71mp1i 13 . . 3 (𝐴𝑉 → (((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆):𝑆1-1-onto→(𝒫 𝐴 ∩ Fin) ↔ 𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin)))
7366, 72bitrd 268 . 2 (𝐴𝑉 → (((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆) ↔ 𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin)))
749, 73mpbid 222 1 (𝐴𝑉𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  cres 5145  cima 5146  Ord word 5760  suc csuc 5763  Fun wfun 5920  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  (class class class)co 6690   supp csupp 7340  1𝑜c1o 7598  2𝑜c2o 7599  𝑚 cmap 7899  Fincfn 7997   finSupp cfsupp 8316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-supp 7341  df-1o 7605  df-2o 7606  df-map 7901  df-fsupp 8317
This theorem is referenced by:  pwfi2en  37984
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