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Theorem enmappw 43460
Description: The set of all mappings from one set to the powerset of the other is equinumerous to the set of all mappings from the second set to the powerset of the first. (Contributed by RP, 27-Apr-2021.)
Assertion
Ref Expression
enmappw ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵m 𝐴) ≈ (𝒫 𝐴m 𝐵))

Proof of Theorem enmappw
StepHypRef Expression
1 enrelmap 43458 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵m 𝐴))
21ensymd 9032 . 2 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵m 𝐴) ≈ 𝒫 (𝐴 × 𝐵))
3 enrelmapr 43459 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴m 𝐵))
4 entr 9033 . 2 (((𝒫 𝐵m 𝐴) ≈ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴m 𝐵)) → (𝒫 𝐵m 𝐴) ≈ (𝒫 𝐴m 𝐵))
52, 3, 4syl2anc 582 1 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵m 𝐴) ≈ (𝒫 𝐴m 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  𝒫 cpw 4606   class class class wbr 5152   × cxp 5680  (class class class)co 7426  m cmap 8851  cen 8967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-1o 8493  df-2o 8494  df-er 8731  df-map 8853  df-en 8971
This theorem is referenced by:  enmappwid  43461
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