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Theorem enrelmap 37311
 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 37320 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 7937 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2 pwen 8018 . . . 4 ((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
31, 2syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
4 xpexg 6858 . . . . 5 ((𝐵𝑊𝐴𝑉) → (𝐵 × 𝐴) ∈ V)
54ancoms 468 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 × 𝐴) ∈ V)
6 pw2eng 7951 . . . 4 ((𝐵 × 𝐴) ∈ V → 𝒫 (𝐵 × 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
75, 6syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐵 × 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
8 entr 7894 . . 3 ((𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴) ∧ 𝒫 (𝐵 × 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴))) → 𝒫 (𝐴 × 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
93, 7, 8syl2anc 691 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
10 pw2eng 7951 . . . . 5 (𝐵𝑊 → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
11 enrefg 7873 . . . . 5 (𝐴𝑉𝐴𝐴)
12 mapen 8009 . . . . 5 ((𝒫 𝐵 ≈ (2𝑜𝑚 𝐵) ∧ 𝐴𝐴) → (𝒫 𝐵𝑚 𝐴) ≈ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴))
1310, 11, 12syl2anr 494 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵𝑚 𝐴) ≈ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴))
14 2on 7455 . . . . 5 2𝑜 ∈ On
15 simpr 476 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
16 simpl 472 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
17 mapxpen 8011 . . . . 5 ((2𝑜 ∈ On ∧ 𝐵𝑊𝐴𝑉) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
1814, 15, 16, 17mp3an2i 1421 . . . 4 ((𝐴𝑉𝐵𝑊) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
19 entr 7894 . . . 4 (((𝒫 𝐵𝑚 𝐴) ≈ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ∧ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴))) → (𝒫 𝐵𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
2013, 18, 19syl2anc 691 . . 3 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
2120ensymd 7893 . 2 ((𝐴𝑉𝐵𝑊) → (2𝑜𝑚 (𝐵 × 𝐴)) ≈ (𝒫 𝐵𝑚 𝐴))
22 entr 7894 . 2 ((𝒫 (𝐴 × 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐴)) ∧ (2𝑜𝑚 (𝐵 × 𝐴)) ≈ (𝒫 𝐵𝑚 𝐴)) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵𝑚 𝐴))
239, 21, 22syl2anc 691 1 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵𝑚 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  Vcvv 3173  𝒫 cpw 4108   class class class wbr 4583   × cxp 5036  Oncon0 5640  (class class class)co 6549  2𝑜c2o 7441   ↑𝑚 cmap 7744   ≈ cen 7838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-1o 7447  df-2o 7448  df-er 7629  df-map 7746  df-en 7842 This theorem is referenced by:  enrelmapr  37312  enmappw  37313
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