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Theorem pwfi2f1o 37143
Description: The pw2f1o 8009 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 2621 . . . . 5 (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))
21pw2f1o2 37082 . . . 4 (𝐴𝑉 → (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})):(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴)
3 f1of1 6093 . . . 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 3666 . . . 4 {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2𝑜𝑚 𝐴)
75, 6eqsstri 3614 . . 3 𝑆 ⊆ (2𝑜𝑚 𝐴)
8 f1ores 6108 . . 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 693 . 2 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆):𝑆1-1-onto→((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆))
10 elmapfun 7825 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → Fun 𝑦)
11 id 22 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → 𝑦 ∈ (2𝑜𝑚 𝐴))
12 0ex 4750 . . . . . . . . . . . . . 14 ∅ ∈ V
1312a1i 11 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → ∅ ∈ V)
1410, 11, 133jca 1240 . . . . . . . . . . . 12 (𝑦 ∈ (2𝑜𝑚 𝐴) → (Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V))
1514adantl 482 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V))
16 funisfsupp 8224 . . . . . . . . . . 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 7823 . . . . . . . . . . . . . 14 (𝑦 ∈ (2𝑜𝑚 𝐴) → 𝑦:𝐴⟶2𝑜)
2019adantl 482 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → 𝑦:𝐴⟶2𝑜)
21 frnsuppeq 7252 . . . . . . . . . . . . 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 7506 . . . . . . . . . . . . . . . 16 2𝑜 = suc 1𝑜
24 df-suc 5688 . . . . . . . . . . . . . . . . 17 suc 1𝑜 = (1𝑜 ∪ {1𝑜})
2524equncomi 3737 . . . . . . . . . . . . . . . 16 suc 1𝑜 = ({1𝑜} ∪ 1𝑜)
2623, 25eqtri 2643 . . . . . . . . . . . . . . 15 2𝑜 = ({1𝑜} ∪ 1𝑜)
27 df1o2 7517 . . . . . . . . . . . . . . . 16 1𝑜 = {∅}
2827eqcomi 2630 . . . . . . . . . . . . . . 15 {∅} = 1𝑜
2926, 28difeq12i 3704 . . . . . . . . . . . . . 14 (2𝑜 ∖ {∅}) = (({1𝑜} ∪ 1𝑜) ∖ 1𝑜)
30 difun2 4020 . . . . . . . . . . . . . . 15 (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = ({1𝑜} ∖ 1𝑜)
31 incom 3783 . . . . . . . . . . . . . . . . 17 ({1𝑜} ∩ 1𝑜) = (1𝑜 ∩ {1𝑜})
32 1on 7512 . . . . . . . . . . . . . . . . . . 19 1𝑜 ∈ On
3332onordi 5791 . . . . . . . . . . . . . . . . . 18 Ord 1𝑜
34 orddisj 5721 . . . . . . . . . . . . . . . . . 18 (Ord 1𝑜 → (1𝑜 ∩ {1𝑜}) = ∅)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 (1𝑜 ∩ {1𝑜}) = ∅
3631, 35eqtri 2643 . . . . . . . . . . . . . . . 16 ({1𝑜} ∩ 1𝑜) = ∅
37 disj3 3993 . . . . . . . . . . . . . . . 16 (({1𝑜} ∩ 1𝑜) = ∅ ↔ {1𝑜} = ({1𝑜} ∖ 1𝑜))
3836, 37mpbi 220 . . . . . . . . . . . . . . 15 {1𝑜} = ({1𝑜} ∖ 1𝑜)
3930, 38eqtr4i 2646 . . . . . . . . . . . . . 14 (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = {1𝑜}
4029, 39eqtri 2643 . . . . . . . . . . . . 13 (2𝑜 ∖ {∅}) = {1𝑜}
4140imaeq2i 5423 . . . . . . . . . . . 12 (𝑦 “ (2𝑜 ∖ {∅})) = (𝑦 “ {1𝑜})
4222, 41syl6eq 2671 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 supp ∅) = (𝑦 “ {1𝑜}))
4342eleq1d 2683 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (𝑦 “ {1𝑜}) ∈ Fin))
44 cnvimass 5444 . . . . . . . . . . . 12 (𝑦 “ {1𝑜}) ⊆ dom 𝑦
45 fdm 6008 . . . . . . . . . . . . 13 (𝑦:𝐴⟶2𝑜 → dom 𝑦 = 𝐴)
4620, 45syl 17 . . . . . . . . . . . 12 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → dom 𝑦 = 𝐴)
4744, 46syl5sseq 3632 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 “ {1𝑜}) ⊆ 𝐴)
4847biantrurd 529 . . . . . . . . . 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 8212 . . . . . . . . 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 3176 . . . . . . 7 (𝐴𝑉 → {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)})
53 cnveq 5256 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
5453imaeq1d 5424 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 “ {1𝑜}) = (𝑦 “ {1𝑜}))
5554cbvmptv 4710 . . . . . . . 8 (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) = (𝑦 ∈ (2𝑜𝑚 𝐴) ↦ (𝑦 “ {1𝑜}))
5655mptpreima 5587 . . . . . . 7 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)}
5752, 5, 563eqtr4g 2680 . . . . . 6 (𝐴𝑉𝑆 = ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)))
5857imaeq2d 5425 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆) = ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))))
59 f1ofo 6101 . . . . . . 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 3811 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
62 foimacnv 6111 . . . . . 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 693 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
6458, 63eqtrd 2655 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
65 f1oeq3 6086 . . . 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 5408 . . . . . 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 2646 . . . 4 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆) = 𝐹
71 f1oeq1 6084 . . . 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 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   class class class wbr 4613  cmpt 4673  ccnv 5073  dom cdm 5074  cres 5076  cima 5077  Ord word 5681  suc csuc 5684  Fun wfun 5841  wf 5843  1-1wf1 5844  ontowfo 5845  1-1-ontowf1o 5846  (class class class)co 6604   supp csupp 7240  1𝑜c1o 7498  2𝑜c2o 7499  𝑚 cmap 7802  Fincfn 7899   finSupp cfsupp 8219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-supp 7241  df-1o 7505  df-2o 7506  df-map 7804  df-fsupp 8220
This theorem is referenced by:  pwfi2en  37144
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