Theorem enrelmap 43987

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 43996 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 9103	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2		pwen 9189	. . . 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 7769	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × 𝐴) ∈ V)
5	4	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × 𝐴) ∈ V)
6		pw2eng 9117	. . . 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 9045	. . 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 584	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2o ↑m (𝐵 × 𝐴)))
10		pw2eng 9117	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ≈ (2o ↑m 𝐵))
11		enrefg 9023	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴)
12		mapen 9180	. . . . 5 ⊢ ((𝒫 𝐵 ≈ (2o ↑m 𝐵) ∧ 𝐴 ≈ 𝐴) → (𝒫 𝐵 ↑m 𝐴) ≈ ((2o ↑m 𝐵) ↑m 𝐴))
13	10, 11, 12	syl2anr 597	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑m 𝐴) ≈ ((2o ↑m 𝐵) ↑m 𝐴))
14		2on 8519	. . . . 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 9182	. . . . 5 ⊢ ((2o ∈ On ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((2o ↑m 𝐵) ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
18	14, 15, 16, 17	mp3an2i 1465	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2o ↑m 𝐵) ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
19		entr 9045	. . . 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 584	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
21	20	ensymd 9044	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2o ↑m (𝐵 × 𝐴)) ≈ (𝒫 𝐵 ↑m 𝐴))
22		entr 9045	. 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 584	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑m 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2106  Vcvv 3478  𝒫 cpw 4605   class class class wbr 5148   × cxp 5687  Oncon0 6386  (class class class)co 7431  2_oc2o 8499   ↑_m cmap 8865   ≈ cen 8981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985

This theorem is referenced by:  enrelmapr  43988  enmappw  43989