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Theorem enrelmap 44010
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. See rfovf1od 44019 for a demonstration of a natural one-to-one onto mapping. (Contributed by RP, 27-Apr-2021.)
Assertion
Ref Expression
enrelmap ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵m 𝐴))

Proof of Theorem enrelmap
StepHypRef Expression
1 xpcomeng 9104 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2 pwen 9190 . . . 4 ((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
31, 2syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
4 xpexg 7770 . . . . 5 ((𝐵𝑊𝐴𝑉) → (𝐵 × 𝐴) ∈ V)
54ancoms 458 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 × 𝐴) ∈ V)
6 pw2eng 9118 . . . 4 ((𝐵 × 𝐴) ∈ V → 𝒫 (𝐵 × 𝐴) ≈ (2om (𝐵 × 𝐴)))
75, 6syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐵 × 𝐴) ≈ (2om (𝐵 × 𝐴)))
8 entr 9046 . . 3 ((𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴) ∧ 𝒫 (𝐵 × 𝐴) ≈ (2om (𝐵 × 𝐴))) → 𝒫 (𝐴 × 𝐵) ≈ (2om (𝐵 × 𝐴)))
93, 7, 8syl2anc 584 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2om (𝐵 × 𝐴)))
10 pw2eng 9118 . . . . 5 (𝐵𝑊 → 𝒫 𝐵 ≈ (2om 𝐵))
11 enrefg 9024 . . . . 5 (𝐴𝑉𝐴𝐴)
12 mapen 9181 . . . . 5 ((𝒫 𝐵 ≈ (2om 𝐵) ∧ 𝐴𝐴) → (𝒫 𝐵m 𝐴) ≈ ((2om 𝐵) ↑m 𝐴))
1310, 11, 12syl2anr 597 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵m 𝐴) ≈ ((2om 𝐵) ↑m 𝐴))
14 2on 8520 . . . . 5 2o ∈ On
15 simpr 484 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
16 simpl 482 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
17 mapxpen 9183 . . . . 5 ((2o ∈ On ∧ 𝐵𝑊𝐴𝑉) → ((2om 𝐵) ↑m 𝐴) ≈ (2om (𝐵 × 𝐴)))
1814, 15, 16, 17mp3an2i 1468 . . . 4 ((𝐴𝑉𝐵𝑊) → ((2om 𝐵) ↑m 𝐴) ≈ (2om (𝐵 × 𝐴)))
19 entr 9046 . . . 4 (((𝒫 𝐵m 𝐴) ≈ ((2om 𝐵) ↑m 𝐴) ∧ ((2om 𝐵) ↑m 𝐴) ≈ (2om (𝐵 × 𝐴))) → (𝒫 𝐵m 𝐴) ≈ (2om (𝐵 × 𝐴)))
2013, 18, 19syl2anc 584 . . 3 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵m 𝐴) ≈ (2om (𝐵 × 𝐴)))
2120ensymd 9045 . 2 ((𝐴𝑉𝐵𝑊) → (2om (𝐵 × 𝐴)) ≈ (𝒫 𝐵m 𝐴))
22 entr 9046 . 2 ((𝒫 (𝐴 × 𝐵) ≈ (2om (𝐵 × 𝐴)) ∧ (2om (𝐵 × 𝐴)) ≈ (𝒫 𝐵m 𝐴)) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵m 𝐴))
239, 21, 22syl2anc 584 1 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  Vcvv 3480  𝒫 cpw 4600   class class class wbr 5143   × cxp 5683  Oncon0 6384  (class class class)co 7431  2oc2o 8500  m cmap 8866  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986
This theorem is referenced by:  enrelmapr  44011  enmappw  44012
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