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Theorem enmappw 40635
 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 40633 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵m 𝐴))
21ensymd 8547 . 2 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵m 𝐴) ≈ 𝒫 (𝐴 × 𝐵))
3 enrelmapr 40634 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴m 𝐵))
4 entr 8548 . 2 (((𝒫 𝐵m 𝐴) ≈ 𝒫 (𝐴 × 𝐵) ∧ 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴m 𝐵)) → (𝒫 𝐵m 𝐴) ≈ (𝒫 𝐴m 𝐵))
52, 3, 4syl2anc 587 1 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵m 𝐴) ≈ (𝒫 𝐴m 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2114  𝒫 cpw 4511   class class class wbr 5042   × cxp 5530  (class class class)co 7140   ↑m cmap 8393   ≈ cen 8493 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-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-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-lim 6174  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-om 7566  df-1st 7675  df-2nd 7676  df-1o 8089  df-2o 8090  df-er 8276  df-map 8395  df-en 8497 This theorem is referenced by:  enmappwid  40636
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