Theorem enrelmap 41223

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 41232 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 8715	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2		pwen 8797	. . . 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 7513	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × 𝐴) ∈ V)
5	4	ancoms 462	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × 𝐴) ∈ V)
6		pw2eng 8729	. . . 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 8658	. . 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 587	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2o ↑m (𝐵 × 𝐴)))
10		pw2eng 8729	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ≈ (2o ↑m 𝐵))
11		enrefg 8638	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴)
12		mapen 8788	. . . . 5 ⊢ ((𝒫 𝐵 ≈ (2o ↑m 𝐵) ∧ 𝐴 ≈ 𝐴) → (𝒫 𝐵 ↑m 𝐴) ≈ ((2o ↑m 𝐵) ↑m 𝐴))
13	10, 11, 12	syl2anr 600	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑m 𝐴) ≈ ((2o ↑m 𝐵) ↑m 𝐴))
14		2on 8188	. . . . 5 ⊢ 2o ∈ On
15		simpr 488	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
16		simpl 486	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
17		mapxpen 8790	. . . . 5 ⊢ ((2o ∈ On ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((2o ↑m 𝐵) ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
18	14, 15, 16, 17	mp3an2i 1468	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2o ↑m 𝐵) ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
19		entr 8658	. . . 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 587	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴)))
21	20	ensymd 8657	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2o ↑m (𝐵 × 𝐴)) ≈ (𝒫 𝐵 ↑m 𝐴))
22		entr 8658	. 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 587	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑m 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2112  Vcvv 3398  𝒫 cpw 4499   class class class wbr 5039   × cxp 5534  Oncon0 6191  (class class class)co 7191  2_oc2o 8174   ↑_m cmap 8486   ≈ cen 8601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-1st 7739  df-2nd 7740  df-1o 8180  df-2o 8181  df-er 8369  df-map 8488  df-en 8605

This theorem is referenced by:  enrelmapr  41224  enmappw  41225