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Theorem pwfi2en 39971
 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 2822 . . 3 (𝑥𝑆 ↦ (𝑥 “ {1o})) = (𝑥𝑆 ↦ (𝑥 “ {1o}))
31, 2pwfi2f1o 39970 . 2 (𝐴𝑉 → (𝑥𝑆 ↦ (𝑥 “ {1o})):𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
4 ovex 7173 . . . 4 (2om 𝐴) ∈ V
51, 4rabex2 5213 . . 3 𝑆 ∈ V
65f1oen 8517 . 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 1538   ∈ wcel 2114  {crab 3134   ∩ cin 3907  ∅c0 4265  𝒫 cpw 4511  {csn 4539   class class class wbr 5042   ↦ cmpt 5122  ◡ccnv 5531   “ cima 5535  –1-1-onto→wf1o 6333  (class class class)co 7140  1oc1o 8082  2oc2o 8083   ↑m cmap 8393   ≈ cen 8493  Fincfn 8496   finSupp cfsupp 8821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ord 6172  df-on 6173  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-supp 7818  df-1o 8089  df-2o 8090  df-map 8395  df-en 8497  df-fsupp 8822 This theorem is referenced by:  frlmpwfi  39972
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