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Theorem pwfi2en 43114
Description: Finitely supported indicator functions are equinumerous to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.)
Hypothesis
Ref Expression
pwfi2en.s 𝑆 = {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅}
Assertion
Ref Expression
pwfi2en (𝐴𝑉𝑆 ≈ (𝒫 𝐴 ∩ Fin))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑉
Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem pwfi2en
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pwfi2en.s . . 3 𝑆 = {𝑦 ∈ (2om 𝐴) ∣ 𝑦 finSupp ∅}
2 eqid 2736 . . 3 (𝑥𝑆 ↦ (𝑥 “ {1o})) = (𝑥𝑆 ↦ (𝑥 “ {1o}))
31, 2pwfi2f1o 43113 . 2 (𝐴𝑉 → (𝑥𝑆 ↦ (𝑥 “ {1o})):𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
4 ovex 7465 . . . 4 (2om 𝐴) ∈ V
51, 4rabex2 5340 . . 3 𝑆 ∈ V
65f1oen 9014 . 2 ((𝑥𝑆 ↦ (𝑥 “ {1o})):𝑆1-1-onto→(𝒫 𝐴 ∩ Fin) → 𝑆 ≈ (𝒫 𝐴 ∩ Fin))
73, 6syl 17 1 (𝐴𝑉𝑆 ≈ (𝒫 𝐴 ∩ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  {crab 3435  cin 3949  c0 4332  𝒫 cpw 4599  {csn 4625   class class class wbr 5142  cmpt 5224  ccnv 5683  cima 5687  1-1-ontowf1o 6559  (class class class)co 7432  1oc1o 8500  2oc2o 8501  m cmap 8867  cen 8983  Fincfn 8986   finSupp cfsupp 9402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-supp 8187  df-1o 8507  df-2o 8508  df-map 8869  df-en 8987  df-fsupp 9403
This theorem is referenced by:  frlmpwfi  43115
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