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Theorem enrelmapr 40692
 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 8596 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2 pwen 8678 . . 3 ((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
31, 2syl 17 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
4 enrelmap 40691 . . 3 ((𝐵𝑊𝐴𝑉) → 𝒫 (𝐵 × 𝐴) ≈ (𝒫 𝐴m 𝐵))
54ancoms 462 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐵 × 𝐴) ≈ (𝒫 𝐴m 𝐵))
6 entr 8548 . 2 ((𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴) ∧ 𝒫 (𝐵 × 𝐴) ≈ (𝒫 𝐴m 𝐵)) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴m 𝐵))
73, 5, 6syl2anc 587 1 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐴m 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2112  𝒫 cpw 4500   class class class wbr 5033   × cxp 5521  (class class class)co 7139   ↑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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  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:  enmappw  40693
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