Theorem enrelmap 39240

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 39249 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 8340	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2		pwen 8421	. . . 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 7237	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × 𝐴) ∈ V)
5	4	ancoms 452	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × 𝐴) ∈ V)
6		pw2eng 8354	. . . 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 8293	. . 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 579	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2o ↑𝑚 (𝐵 × 𝐴)))
10		pw2eng 8354	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ≈ (2o ↑𝑚 𝐵))
11		enrefg 8273	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴)
12		mapen 8412	. . . . 5 ⊢ ((𝒫 𝐵 ≈ (2o ↑𝑚 𝐵) ∧ 𝐴 ≈ 𝐴) → (𝒫 𝐵 ↑𝑚 𝐴) ≈ ((2o ↑𝑚 𝐵) ↑𝑚 𝐴))
13	10, 11, 12	syl2anr 590	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑𝑚 𝐴) ≈ ((2o ↑𝑚 𝐵) ↑𝑚 𝐴))
14		2on 7852	. . . . 5 ⊢ 2o ∈ On
15		simpr 479	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
16		simpl 476	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
17		mapxpen 8414	. . . . 5 ⊢ ((2o ∈ On ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((2o ↑𝑚 𝐵) ↑𝑚 𝐴) ≈ (2o ↑𝑚 (𝐵 × 𝐴)))
18	14, 15, 16, 17	mp3an2i 1539	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2o ↑𝑚 𝐵) ↑𝑚 𝐴) ≈ (2o ↑𝑚 (𝐵 × 𝐴)))
19		entr 8293	. . . 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 579	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑𝑚 𝐴) ≈ (2o ↑𝑚 (𝐵 × 𝐴)))
21	20	ensymd 8292	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2o ↑𝑚 (𝐵 × 𝐴)) ≈ (𝒫 𝐵 ↑𝑚 𝐴))
22		entr 8293	. 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 579	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑𝑚 𝐴))

Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2106  Vcvv 3397  𝒫 cpw 4378   class class class wbr 4886   × cxp 5353  Oncon0 5976  (class class class)co 6922  2_oc2o 7837   ↑_𝑚 cmap 8140   ≈ cen 8238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-1o 7843  df-2o 7844  df-er 8026  df-map 8142  df-en 8242

This theorem is referenced by:  enrelmapr  39241  enmappw  39242