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Theorem enrelmapr 39127
Description: The set of all possible relations between two sets is equinumerous to the set of all mappings from one set to the powerset of the other. (Contributed by RP, 27-Apr-2021.)
Assertion
Ref Expression
enrelmapr ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴𝑚 𝐵))

Proof of Theorem enrelmapr
StepHypRef Expression
1 xpcomeng 8327 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2 pwen 8408 . . 3 ((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
31, 2syl 17 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
4 enrelmap 39126 . . 3 ((𝐵𝑊𝐴𝑉) → 𝒫 (𝐵 × 𝐴) ≈ (𝒫 𝐴𝑚 𝐵))
54ancoms 452 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐵 × 𝐴) ≈ (𝒫 𝐴𝑚 𝐵))
6 entr 8280 . 2 ((𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴) ∧ 𝒫 (𝐵 × 𝐴) ≈ (𝒫 𝐴𝑚 𝐵)) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴𝑚 𝐵))
73, 5, 6syl2anc 579 1 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴𝑚 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2164  𝒫 cpw 4380   class class class wbr 4875   × cxp 5344  (class class class)co 6910  𝑚 cmap 8127  cen 8225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-1o 7831  df-2o 7832  df-er 8014  df-map 8129  df-en 8229
This theorem is referenced by:  enmappw  39128
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