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Theorem pwfi2f1o 36684
Description: The pw2f1o 7950 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 2610 . . . . 5 (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) = (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜}))
21pw2f1o2 36623 . . . 4 (𝐴𝑉 → (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})):(2𝑜𝑚 𝐴)–1-1-onto→𝒫 𝐴)
3 f1of1 6049 . . . 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 3650 . . . 4 {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2𝑜𝑚 𝐴)
75, 6eqsstri 3598 . . 3 𝑆 ⊆ (2𝑜𝑚 𝐴)
8 f1ores 6064 . . 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 7767 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → Fun 𝑦)
11 id 22 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → 𝑦 ∈ (2𝑜𝑚 𝐴))
12 0ex 4718 . . . . . . . . . . . . . 14 ∅ ∈ V
1312a1i 11 . . . . . . . . . . . . 13 (𝑦 ∈ (2𝑜𝑚 𝐴) → ∅ ∈ V)
1410, 11, 133jca 1235 . . . . . . . . . . . 12 (𝑦 ∈ (2𝑜𝑚 𝐴) → (Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V))
1514adantl 481 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V))
16 funisfsupp 8163 . . . . . . . . . . 11 ((Fun 𝑦𝑦 ∈ (2𝑜𝑚 𝐴) ∧ ∅ ∈ V) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1715, 16syl 17 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
1813anim2i 591 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝐴𝑉 ∧ ∅ ∈ V))
19 elmapi 7765 . . . . . . . . . . . . . 14 (𝑦 ∈ (2𝑜𝑚 𝐴) → 𝑦:𝐴⟶2𝑜)
2019adantl 481 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → 𝑦:𝐴⟶2𝑜)
21 frnsuppeq 7194 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ ∅ ∈ V) → (𝑦:𝐴⟶2𝑜 → (𝑦 supp ∅) = (𝑦 “ (2𝑜 ∖ {∅}))))
2218, 20, 21sylc 63 . . . . . . . . . . . 12 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 supp ∅) = (𝑦 “ (2𝑜 ∖ {∅})))
23 df-2o 7448 . . . . . . . . . . . . . . . 16 2𝑜 = suc 1𝑜
24 df-suc 5646 . . . . . . . . . . . . . . . . 17 suc 1𝑜 = (1𝑜 ∪ {1𝑜})
2524equncomi 3721 . . . . . . . . . . . . . . . 16 suc 1𝑜 = ({1𝑜} ∪ 1𝑜)
2623, 25eqtri 2632 . . . . . . . . . . . . . . 15 2𝑜 = ({1𝑜} ∪ 1𝑜)
27 df1o2 7459 . . . . . . . . . . . . . . . 16 1𝑜 = {∅}
2827eqcomi 2619 . . . . . . . . . . . . . . 15 {∅} = 1𝑜
2926, 28difeq12i 3688 . . . . . . . . . . . . . 14 (2𝑜 ∖ {∅}) = (({1𝑜} ∪ 1𝑜) ∖ 1𝑜)
30 difun2 4000 . . . . . . . . . . . . . . 15 (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = ({1𝑜} ∖ 1𝑜)
31 incom 3767 . . . . . . . . . . . . . . . . 17 ({1𝑜} ∩ 1𝑜) = (1𝑜 ∩ {1𝑜})
32 1on 7454 . . . . . . . . . . . . . . . . . . 19 1𝑜 ∈ On
3332onordi 5749 . . . . . . . . . . . . . . . . . 18 Ord 1𝑜
34 orddisj 5679 . . . . . . . . . . . . . . . . . 18 (Ord 1𝑜 → (1𝑜 ∩ {1𝑜}) = ∅)
3533, 34ax-mp 5 . . . . . . . . . . . . . . . . 17 (1𝑜 ∩ {1𝑜}) = ∅
3631, 35eqtri 2632 . . . . . . . . . . . . . . . 16 ({1𝑜} ∩ 1𝑜) = ∅
37 disj3 3973 . . . . . . . . . . . . . . . 16 (({1𝑜} ∩ 1𝑜) = ∅ ↔ {1𝑜} = ({1𝑜} ∖ 1𝑜))
3836, 37mpbi 219 . . . . . . . . . . . . . . 15 {1𝑜} = ({1𝑜} ∖ 1𝑜)
3930, 38eqtr4i 2635 . . . . . . . . . . . . . 14 (({1𝑜} ∪ 1𝑜) ∖ 1𝑜) = {1𝑜}
4029, 39eqtri 2632 . . . . . . . . . . . . 13 (2𝑜 ∖ {∅}) = {1𝑜}
4140imaeq2i 5383 . . . . . . . . . . . 12 (𝑦 “ (2𝑜 ∖ {∅})) = (𝑦 “ {1𝑜})
4222, 41syl6eq 2660 . . . . . . . . . . 11 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 supp ∅) = (𝑦 “ {1𝑜}))
4342eleq1d 2672 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (𝑦 “ {1𝑜}) ∈ Fin))
44 cnvimass 5404 . . . . . . . . . . . 12 (𝑦 “ {1𝑜}) ⊆ dom 𝑦
45 fdm 5964 . . . . . . . . . . . . 13 (𝑦:𝐴⟶2𝑜 → dom 𝑦 = 𝐴)
4620, 45syl 17 . . . . . . . . . . . 12 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → dom 𝑦 = 𝐴)
4744, 46syl5sseq 3616 . . . . . . . . . . 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 293 . . . . . . . . 9 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ ((𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (𝑦 “ {1𝑜}) ∈ Fin)))
50 elfpw 8151 . . . . . . . . 9 ((𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((𝑦 “ {1𝑜}) ⊆ 𝐴 ∧ (𝑦 “ {1𝑜}) ∈ Fin))
5149, 50syl6bbr 277 . . . . . . . 8 ((𝐴𝑉𝑦 ∈ (2𝑜𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)))
5251rabbidva 3163 . . . . . . 7 (𝐴𝑉 → {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)})
53 cnveq 5218 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
5453imaeq1d 5384 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 “ {1𝑜}) = (𝑦 “ {1𝑜}))
5554cbvmptv 4678 . . . . . . . 8 (𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) = (𝑦 ∈ (2𝑜𝑚 𝐴) ↦ (𝑦 “ {1𝑜}))
5655mptpreima 5545 . . . . . . 7 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2𝑜𝑚 𝐴) ∣ (𝑦 “ {1𝑜}) ∈ (𝒫 𝐴 ∩ Fin)}
5752, 5, 563eqtr4g 2669 . . . . . 6 (𝐴𝑉𝑆 = ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin)))
5857imaeq2d 5385 . . . . 5 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆) = ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ (𝒫 𝐴 ∩ Fin))))
59 f1ofo 6057 . . . . . . 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 3795 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
62 foimacnv 6067 . . . . . 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 2644 . . . 4 (𝐴𝑉 → ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
65 f1oeq3 6042 . . . 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 5369 . . . . . 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 2635 . . . 4 ((𝑥 ∈ (2𝑜𝑚 𝐴) ↦ (𝑥 “ {1𝑜})) ↾ 𝑆) = 𝐹
71 f1oeq1 6040 . . . 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 267 . 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 221 1 (𝐴𝑉𝐹:𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  {crab 2900  Vcvv 3173  cdif 3537  cun 3538  cin 3539  wss 3540  c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  cmpt 4643  ccnv 5037  dom cdm 5038  cres 5040  cima 5041  Ord word 5639  suc csuc 5642  Fun wfun 5798  wf 5800  1-1wf1 5801  ontowfo 5802  1-1-ontowf1o 5803  (class class class)co 6549   supp csupp 7182  1𝑜c1o 7440  2𝑜c2o 7441  𝑚 cmap 7744  Fincfn 7841   finSupp cfsupp 8158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-supp 7183  df-1o 7447  df-2o 7448  df-map 7746  df-fsupp 8159
This theorem is referenced by:  pwfi2en  36685
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