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Theorem pwfi2en 37486
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 2620 . . 3 (𝑥𝑆 ↦ (𝑥 “ {1𝑜})) = (𝑥𝑆 ↦ (𝑥 “ {1𝑜}))
31, 2pwfi2f1o 37485 . 2 (𝐴𝑉 → (𝑥𝑆 ↦ (𝑥 “ {1𝑜})):𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
4 ovex 6663 . . . 4 (2𝑜𝑚 𝐴) ∈ V
51, 4rabex2 4806 . . 3 𝑆 ∈ V
65f1oen 7961 . 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 1481  wcel 1988  {crab 2913  cin 3566  c0 3907  𝒫 cpw 4149  {csn 4168   class class class wbr 4644  cmpt 4720  ccnv 5103  cima 5107  1-1-ontowf1o 5875  (class class class)co 6635  1𝑜c1o 7538  2𝑜c2o 7539  𝑚 cmap 7842  cen 7937  Fincfn 7940   finSupp cfsupp 8260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-supp 7281  df-1o 7545  df-2o 7546  df-map 7844  df-en 7941  df-fsupp 8261
This theorem is referenced by:  frlmpwfi  37487
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