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Theorem pwfi2f1o 38192
Description: The pw2f1o 8221 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 2771 . . . . 5 (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))
21pw2f1o2 38131 . . . 4 (𝐴𝑉 → (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})):(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴)
3 f1of1 6277 . . . 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 3836 . . . 4 {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2𝑜𝑚 𝐴)
75, 6eqsstri 3784 . . 3 𝑆 ⊆ (2𝑜𝑚 𝐴)
8 f1ores 6292 . . 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 574 . 2 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆))
10 elmapfun 8033 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → Fun 𝑦)
11 id 22 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → 𝑦 ∈ (2𝑜𝑚 𝐴))
12 0ex 4924 . . . . . . . . . . . . . 14 ∅ ∈ V
1312a1i 11 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → ∅ ∈ V)
1410, 11, 133jca 1122 . . . . . . . . . . . 12 (𝑦 ∈ (2𝑜𝑚 𝐴) → (Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V))
1514adantl 467 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V))
16 funisfsupp 8436 . . . . . . . . . . 11 ((Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1715, 16syl 17 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1813anim2i 603 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝐴𝑉 ∧ ∅ ∈ V))
19 elmapi 8031 . . . . . . . . . . . . . 14 (𝑦 ∈ (2𝑜𝑚 𝐴) → 𝑦:𝐴⟶2𝑜)
2019adantl 467 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → 𝑦:𝐴⟶2𝑜)
21 frnsuppeq 7458 . . . . . . . . . . . . 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 7714 . . . . . . . . . . . . . . . 16 2𝑜 = suc 1𝑜
24 df-suc 5872 . . . . . . . . . . . . . . . . 17 suc 1𝑜 = (1𝑜 ∪ {1𝑜})
2524equncomi 3910 . . . . . . . . . . . . . . . 16 suc 1𝑜 = ({1𝑜} ∪ 1𝑜)
2623, 25eqtri 2793 . . . . . . . . . . . . . . 15 2𝑜 = ({1𝑜} ∪ 1𝑜)
27 df1o2 7726 . . . . . . . . . . . . . . . 16 1𝑜 = {∅}
2827eqcomi 2780 . . . . . . . . . . . . . . 15 {∅} = 1𝑜
2926, 28difeq12i 3877 . . . . . . . . . . . . . 14 (2𝑜 ∖ {∅}) = (({1𝑜} ∪ 1𝑜) ∖ 1𝑜)
30 difun2 4190 . . . . . . . . . . . . . . 15 (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = ({1𝑜} ∖ 1𝑜)
31 incom 3956 . . . . . . . . . . . . . . . . 17 ({1𝑜} ∩ 1𝑜) = (1𝑜 ∩ {1𝑜})
32 1on 7720 . . . . . . . . . . . . . . . . . . 19 1𝑜 ∈ On
3332onordi 5975 . . . . . . . . . . . . . . . . . 18 Ord 1𝑜
34 orddisj 5905 . . . . . . . . . . . . . . . . . 18 (Ord 1𝑜 → (1𝑜 ∩ {1𝑜}) = ∅)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 (1𝑜 ∩ {1𝑜}) = ∅
3631, 35eqtri 2793 . . . . . . . . . . . . . . . 16 ({1𝑜} ∩ 1𝑜) = ∅
37 disj3 4164 . . . . . . . . . . . . . . . 16 (({1𝑜} ∩ 1𝑜) = ∅ ↔ {1𝑜} = ({1𝑜} ∖ 1𝑜))
3836, 37mpbi 220 . . . . . . . . . . . . . . 15 {1𝑜} = ({1𝑜} ∖ 1𝑜)
3930, 38eqtr4i 2796 . . . . . . . . . . . . . 14 (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = {1𝑜}
4029, 39eqtri 2793 . . . . . . . . . . . . 13 (2𝑜 ∖ {∅}) = {1𝑜}
4140imaeq2i 5605 . . . . . . . . . . . 12 (𝑦 “ (2𝑜 ∖ {∅})) = (𝑦 “ {1𝑜})
4222, 41syl6eq 2821 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 supp ∅) = (𝑦 “ {1𝑜}))
4342eleq1d 2835 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (𝑦 “ {1𝑜}) ∈ Fin))
44 cnvimass 5626 . . . . . . . . . . . 12 (𝑦 “ {1𝑜}) ⊆ dom 𝑦
45 fdm 6191 . . . . . . . . . . . . 13 (𝑦:𝐴⟶2𝑜 → dom 𝑦 = 𝐴)
4620, 45syl 17 . . . . . . . . . . . 12 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → dom 𝑦 = 𝐴)
4744, 46syl5sseq 3802 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 “ {1𝑜}) ⊆ 𝐴)
4847biantrurd 522 . . . . . . . . . 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 8424 . . . . . . . . 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 3338 . . . . . . 7 (𝐴𝑉 → {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)})
53 cnveq 5434 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
5453imaeq1d 5606 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 “ {1𝑜}) = (𝑦 “ {1𝑜}))
5554cbvmptv 4884 . . . . . . . 8 (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) = (𝑦 ∈ (2𝑜𝑚 𝐴) ↦ (𝑦 “ {1𝑜}))
5655mptpreima 5772 . . . . . . 7 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)}
5752, 5, 563eqtr4g 2830 . . . . . 6 (𝐴𝑉𝑆 = ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)))
5857imaeq2d 5607 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆) = ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))))
59 f1ofo 6285 . . . . . . 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 3981 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
62 foimacnv 6295 . . . . . 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 574 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
6458, 63eqtrd 2805 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
65 f1oeq3 6270 . . . 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 5590 . . . . . 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 2796 . . . 4 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆) = 𝐹
71 f1oeq1 6268 . . . 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 382  w3a 1071   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  {csn 4316   class class class wbr 4786  cmpt 4863  ccnv 5248  dom cdm 5249  cres 5251  cima 5252  Ord word 5865  suc csuc 5868  Fun wfun 6025  wf 6027  1-1wf1 6028  ontowfo 6029  1-1-ontowf1o 6030  (class class class)co 6793   supp csupp 7446  1𝑜c1o 7706  2𝑜c2o 7707  𝑚 cmap 8009  Fincfn 8109   finSupp cfsupp 8431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-supp 7447  df-1o 7713  df-2o 7714  df-map 8011  df-fsupp 8432
This theorem is referenced by:  pwfi2en  38193
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