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Theorem pwfi2en 41453
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 2737 . . 3 (𝑥𝑆 ↦ (𝑥 “ {1o})) = (𝑥𝑆 ↦ (𝑥 “ {1o}))
31, 2pwfi2f1o 41452 . 2 (𝐴𝑉 → (𝑥𝑆 ↦ (𝑥 “ {1o})):𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
4 ovex 7395 . . . 4 (2om 𝐴) ∈ V
51, 4rabex2 5296 . . 3 𝑆 ∈ V
65f1oen 8920 . 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 1542  wcel 2107  {crab 3410  cin 3914  c0 4287  𝒫 cpw 4565  {csn 4591   class class class wbr 5110  cmpt 5193  ccnv 5637  cima 5641  1-1-ontowf1o 6500  (class class class)co 7362  1oc1o 8410  2oc2o 8411  m cmap 8772  cen 8887  Fincfn 8890   finSupp cfsupp 9312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-supp 8098  df-1o 8417  df-2o 8418  df-map 8774  df-en 8891  df-fsupp 9313
This theorem is referenced by:  frlmpwfi  41454
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