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Theorem enrelmap 43492
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 43501 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 9087 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2 pwen 9173 . . . 4 ((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
31, 2syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
4 xpexg 7750 . . . . 5 ((𝐵𝑊𝐴𝑉) → (𝐵 × 𝐴) ∈ V)
54ancoms 457 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 × 𝐴) ∈ V)
6 pw2eng 9101 . . . 4 ((𝐵 × 𝐴) ∈ V → 𝒫 (𝐵 × 𝐴) ≈ (2om (𝐵 × 𝐴)))
75, 6syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐵 × 𝐴) ≈ (2om (𝐵 × 𝐴)))
8 entr 9025 . . 3 ((𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴) ∧ 𝒫 (𝐵 × 𝐴) ≈ (2om (𝐵 × 𝐴))) → 𝒫 (𝐴 × 𝐵) ≈ (2om (𝐵 × 𝐴)))
93, 7, 8syl2anc 582 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2om (𝐵 × 𝐴)))
10 pw2eng 9101 . . . . 5 (𝐵𝑊 → 𝒫 𝐵 ≈ (2om 𝐵))
11 enrefg 9003 . . . . 5 (𝐴𝑉𝐴𝐴)
12 mapen 9164 . . . . 5 ((𝒫 𝐵 ≈ (2om 𝐵) ∧ 𝐴𝐴) → (𝒫 𝐵m 𝐴) ≈ ((2om 𝐵) ↑m 𝐴))
1310, 11, 12syl2anr 595 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵m 𝐴) ≈ ((2om 𝐵) ↑m 𝐴))
14 2on 8499 . . . . 5 2o ∈ On
15 simpr 483 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
16 simpl 481 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
17 mapxpen 9166 . . . . 5 ((2o ∈ On ∧ 𝐵𝑊𝐴𝑉) → ((2om 𝐵) ↑m 𝐴) ≈ (2om (𝐵 × 𝐴)))
1814, 15, 16, 17mp3an2i 1462 . . . 4 ((𝐴𝑉𝐵𝑊) → ((2om 𝐵) ↑m 𝐴) ≈ (2om (𝐵 × 𝐴)))
19 entr 9025 . . . 4 (((𝒫 𝐵m 𝐴) ≈ ((2om 𝐵) ↑m 𝐴) ∧ ((2om 𝐵) ↑m 𝐴) ≈ (2om (𝐵 × 𝐴))) → (𝒫 𝐵m 𝐴) ≈ (2om (𝐵 × 𝐴)))
2013, 18, 19syl2anc 582 . . 3 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵m 𝐴) ≈ (2om (𝐵 × 𝐴)))
2120ensymd 9024 . 2 ((𝐴𝑉𝐵𝑊) → (2om (𝐵 × 𝐴)) ≈ (𝒫 𝐵m 𝐴))
22 entr 9025 . 2 ((𝒫 (𝐴 × 𝐵) ≈ (2om (𝐵 × 𝐴)) ∧ (2om (𝐵 × 𝐴)) ≈ (𝒫 𝐵m 𝐴)) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵m 𝐴))
239, 21, 22syl2anc 582 1 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  Vcvv 3463  𝒫 cpw 4598   class class class wbr 5143   × cxp 5670  Oncon0 6364  (class class class)co 7416  2oc2o 8479  m cmap 8843  cen 8959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-1o 8485  df-2o 8486  df-er 8723  df-map 8845  df-en 8963
This theorem is referenced by:  enrelmapr  43493  enmappw  43494
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