Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwfi2en Structured version   Visualization version   GIF version

Theorem pwfi2en 43086
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 2735 . . 3 (𝑥𝑆 ↦ (𝑥 “ {1o})) = (𝑥𝑆 ↦ (𝑥 “ {1o}))
31, 2pwfi2f1o 43085 . 2 (𝐴𝑉 → (𝑥𝑆 ↦ (𝑥 “ {1o})):𝑆1-1-onto→(𝒫 𝐴 ∩ Fin))
4 ovex 7464 . . . 4 (2om 𝐴) ∈ V
51, 4rabex2 5347 . . 3 𝑆 ∈ V
65f1oen 9012 . 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 1537  wcel 2106  {crab 3433  cin 3962  c0 4339  𝒫 cpw 4605  {csn 4631   class class class wbr 5148  cmpt 5231  ccnv 5688  cima 5692  1-1-ontowf1o 6562  (class class class)co 7431  1oc1o 8498  2oc2o 8499  m cmap 8865  cen 8981  Fincfn 8984   finSupp cfsupp 9399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-supp 8185  df-1o 8505  df-2o 8506  df-map 8867  df-en 8985  df-fsupp 9400
This theorem is referenced by:  frlmpwfi  43087
  Copyright terms: Public domain W3C validator