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Theorem enmappw 40223
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 40221 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵m 𝐴))
21ensymd 8548 . 2 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵m 𝐴) ≈ 𝒫 (𝐴 × 𝐵))
3 enrelmapr 40222 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴m 𝐵))
4 entr 8549 . 2 (((𝒫 𝐵m 𝐴) ≈ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴m 𝐵)) → (𝒫 𝐵m 𝐴) ≈ (𝒫 𝐴m 𝐵))
52, 3, 4syl2anc 584 1 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵m 𝐴) ≈ (𝒫 𝐴m 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  𝒫 cpw 4535   class class class wbr 5057   × cxp 5546  (class class class)co 7145  m cmap 8395  cen 8494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-1o 8091  df-2o 8092  df-er 8278  df-map 8397  df-en 8498
This theorem is referenced by:  enmappwid  40224
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