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Theorem enrelmap 37208
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 37217 for a demonstration of an 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 7813 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2 pwen 7894 . . . 4 ((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
31, 2syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
4 xpexg 6734 . . . . 5 ((𝐵𝑊𝐴𝑉) → (𝐵 × 𝐴) ∈ V)
54ancoms 467 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 × 𝐴) ∈ V)
6 pw2eng 7827 . . . 4 ((𝐵 × 𝐴) ∈ V → 𝒫 (𝐵 × 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
75, 6syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐵 × 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
8 entr 7770 . . 3 ((𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴) ∧ 𝒫 (𝐵 × 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴))) → 𝒫 (𝐴 × 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
93, 7, 8syl2anc 690 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
10 pw2eng 7827 . . . . 5 (𝐵𝑊 → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
11 enrefg 7749 . . . . 5 (𝐴𝑉𝐴𝐴)
12 mapen 7885 . . . . 5 ((𝒫 𝐵 ≈ (2𝑜𝑚 𝐵) ∧ 𝐴𝐴) → (𝒫 𝐵𝑚 𝐴) ≈ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴))
1310, 11, 12syl2anr 493 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵𝑚 𝐴) ≈ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴))
14 2on 7331 . . . . 5 2𝑜 ∈ On
15 simpr 475 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
16 simpl 471 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
17 mapxpen 7887 . . . . 5 ((2𝑜 ∈ On ∧ 𝐵𝑊𝐴𝑉) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
1814, 15, 16, 17mp3an2i 1420 . . . 4 ((𝐴𝑉𝐵𝑊) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
19 entr 7770 . . . 4 (((𝒫 𝐵𝑚 𝐴) ≈ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ∧ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴))) → (𝒫 𝐵𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
2013, 18, 19syl2anc 690 . . 3 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
2120ensymd 7769 . 2 ((𝐴𝑉𝐵𝑊) → (2𝑜𝑚 (𝐵 × 𝐴)) ≈ (𝒫 𝐵𝑚 𝐴))
22 entr 7770 . 2 ((𝒫 (𝐴 × 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐴)) ∧ (2𝑜𝑚 (𝐵 × 𝐴)) ≈ (𝒫 𝐵𝑚 𝐴)) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵𝑚 𝐴))
239, 21, 22syl2anc 690 1 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1938  Vcvv 3077  𝒫 cpw 4011   class class class wbr 4481   × cxp 4930  Oncon0 5530  (class class class)co 6426  2𝑜c2o 7317  𝑚 cmap 7620  cen 7714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-om 6834  df-1st 6934  df-2nd 6935  df-1o 7323  df-2o 7324  df-er 7505  df-map 7622  df-en 7718
This theorem is referenced by:  enrelmapr  37209  enmappw  37210
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