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Theorem enrelmap 38117
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 38126 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 8049 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2 pwen 8130 . . . 4 ((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
31, 2syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
4 xpexg 6957 . . . . 5 ((𝐵𝑊𝐴𝑉) → (𝐵 × 𝐴) ∈ V)
54ancoms 469 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐵 × 𝐴) ∈ V)
6 pw2eng 8063 . . . 4 ((𝐵 × 𝐴) ∈ V → 𝒫 (𝐵 × 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
75, 6syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐵 × 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
8 entr 8005 . . 3 ((𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴) ∧ 𝒫 (𝐵 × 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴))) → 𝒫 (𝐴 × 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
93, 7, 8syl2anc 693 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
10 pw2eng 8063 . . . . 5 (𝐵𝑊 → 𝒫 𝐵 ≈ (2𝑜𝑚 𝐵))
11 enrefg 7984 . . . . 5 (𝐴𝑉𝐴𝐴)
12 mapen 8121 . . . . 5 ((𝒫 𝐵 ≈ (2𝑜𝑚 𝐵) ∧ 𝐴𝐴) → (𝒫 𝐵𝑚 𝐴) ≈ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴))
1310, 11, 12syl2anr 495 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵𝑚 𝐴) ≈ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴))
14 2on 7565 . . . . 5 2𝑜 ∈ On
15 simpr 477 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
16 simpl 473 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
17 mapxpen 8123 . . . . 5 ((2𝑜 ∈ On ∧ 𝐵𝑊𝐴𝑉) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
1814, 15, 16, 17mp3an2i 1428 . . . 4 ((𝐴𝑉𝐵𝑊) → ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
19 entr 8005 . . . 4 (((𝒫 𝐵𝑚 𝐴) ≈ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ∧ ((2𝑜𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴))) → (𝒫 𝐵𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
2013, 18, 19syl2anc 693 . . 3 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐵𝑚 𝐴) ≈ (2𝑜𝑚 (𝐵 × 𝐴)))
2120ensymd 8004 . 2 ((𝐴𝑉𝐵𝑊) → (2𝑜𝑚 (𝐵 × 𝐴)) ≈ (𝒫 𝐵𝑚 𝐴))
22 entr 8005 . 2 ((𝒫 (𝐴 × 𝐵) ≈ (2𝑜𝑚 (𝐵 × 𝐴)) ∧ (2𝑜𝑚 (𝐵 × 𝐴)) ≈ (𝒫 𝐵𝑚 𝐴)) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵𝑚 𝐴))
239, 21, 22syl2anc 693 1 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1989  Vcvv 3198  𝒫 cpw 4156   class class class wbr 4651   × cxp 5110  Oncon0 5721  (class class class)co 6647  2𝑜c2o 7551  𝑚 cmap 7854  cen 7949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-1o 7557  df-2o 7558  df-er 7739  df-map 7856  df-en 7953
This theorem is referenced by:  enrelmapr  38118  enmappw  38119
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