Theorem pwfi2en 43054

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 ∈ (2o ↑m 𝐴) ∣ 𝑦 finSupp ∅}

Assertion

Ref	Expression
pwfi2en	⊢ (𝐴 ∈ 𝑉 → 𝑆 ≈ (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.

Step	Hyp	Ref	Expression
1		pwfi2en.s	. . 3 ⊢ 𝑆 = {𝑦 ∈ (2o ↑m 𝐴) ∣ 𝑦 finSupp ∅}
2		eqid 2740	. . 3 ⊢ (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1o})) = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1o}))
3	1, 2	pwfi2f1o 43053	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1o})):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin))
4		ovex 7481	. . . 4 ⊢ (2o ↑m 𝐴) ∈ V
5	1, 4	rabex2 5359	. . 3 ⊢ 𝑆 ∈ V
6	5	f1oen 9033	. 2 ⊢ ((𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1o})):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin) → 𝑆 ≈ (𝒫 𝐴 ∩ Fin))
7	3, 6	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → 𝑆 ≈ (𝒫 𝐴 ∩ Fin))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  {crab 3443   ∩ cin 3975  ∅c0 4352  𝒫 cpw 4622  {csn 4648   class class class wbr 5166   ↦ cmpt 5249  ^◡ccnv 5699   “ cima 5703  –1-1-onto→wf1o 6572  (class class class)co 7448  1_oc1o 8515  2_oc2o 8516   ↑_m cmap 8884   ≈ cen 9000  Fincfn 9003   finSupp cfsupp 9431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-supp 8202  df-1o 8522  df-2o 8523  df-map 8886  df-en 9004  df-fsupp 9432

This theorem is referenced by:  frlmpwfi  43055