Theorem enrelmap 43324

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 43333 for a demonstration of a natural one-to-one onto mapping. (Contributed by RP, 27-Apr-2021.)

Assertion

Ref	Expression
enrelmap	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑m 𝐴))

Proof of Theorem enrelmap

Step	Hyp	Ref	Expression
1		xpcomeng 9066	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2		pwen 9152	. . . 4 ⊢ ((𝐴 × 𝐵) ≈ (𝐵 × 𝐴) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
3	1, 2	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴))
4		xpexg 7734	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × 𝐴) ∈ V)
5	4	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × 𝐴) ∈ V)
6		pw2eng 9080	. . . 4 ⊢ ((𝐵 × 𝐴) ∈ V → 𝒫 (𝐵 × 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
7	5, 6	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐵 × 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
8		entr 9004	. . 3 ⊢ ((𝒫 (𝐴 × 𝐵) ≈ 𝒫 (𝐵 × 𝐴) ∧ 𝒫 (𝐵 × 𝐴) ≈ (2o ↑m (𝐵 × 𝐴))) → 𝒫 (𝐴 × 𝐵) ≈ (2o ↑m (𝐵 × 𝐴)))
9	3, 7, 8	syl2anc 583	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2o ↑m (𝐵 × 𝐴)))
10		pw2eng 9080	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ≈ (2o ↑m 𝐵))
11		enrefg 8982	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴)
12		mapen 9143	. . . . 5 ⊢ ((𝒫 𝐵 ≈ (2o ↑m 𝐵) ∧ 𝐴 ≈ 𝐴) → (𝒫 𝐵 ↑m 𝐴) ≈ ((2o ↑m 𝐵) ↑m 𝐴))
13	10, 11, 12	syl2anr 596	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑m 𝐴) ≈ ((2o ↑m 𝐵) ↑m 𝐴))
14		2on 8481	. . . . 5 ⊢ 2o ∈ On
15		simpr 484	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
16		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
17		mapxpen 9145	. . . . 5 ⊢ ((2o ∈ On ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((2o ↑m 𝐵) ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
18	14, 15, 16, 17	mp3an2i 1462	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2o ↑m 𝐵) ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
19		entr 9004	. . . 4 ⊢ (((𝒫 𝐵 ↑m 𝐴) ≈ ((2o ↑m 𝐵) ↑m 𝐴) ∧ ((2o ↑m 𝐵) ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴))) → (𝒫 𝐵 ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
20	13, 18, 19	syl2anc 583	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
21	20	ensymd 9003	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2o ↑m (𝐵 × 𝐴)) ≈ (𝒫 𝐵 ↑m 𝐴))
22		entr 9004	. 2 ⊢ ((𝒫 (𝐴 × 𝐵) ≈ (2o ↑m (𝐵 × 𝐴)) ∧ (2o ↑m (𝐵 × 𝐴)) ≈ (𝒫 𝐵 ↑m 𝐴)) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑m 𝐴))
23	9, 21, 22	syl2anc 583	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑m 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2098  Vcvv 3468  𝒫 cpw 4597   class class class wbr 5141   × cxp 5667  Oncon0 6358  (class class class)co 7405  2_oc2o 8461   ↑_m cmap 8822   ≈ cen 8938

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-1o 8467  df-2o 8468  df-er 8705  df-map 8824  df-en 8942

This theorem is referenced by:  enrelmapr  43325  enmappw  43326