Theorem enrelmap 38291

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 38300 for a demonstration of an 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 8052	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
2		pwen 8133	. . . 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 6960	. . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × 𝐴) ∈ V)
5	4	ancoms 469	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × 𝐴) ∈ V)
6		pw2eng 8066	. . . 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 8008	. . 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 693	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐴)))
10		pw2eng 8066	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵))
11		enrefg 7987	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴)
12		mapen 8124	. . . . 5 ⊢ ((𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵) ∧ 𝐴 ≈ 𝐴) → (𝒫 𝐵 ↑𝑚 𝐴) ≈ ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐴))
13	10, 11, 12	syl2anr 495	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑𝑚 𝐴) ≈ ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐴))
14		2on 7568	. . . . 5 ⊢ 2𝑜 ∈ On
15		simpr 477	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊)
16		simpl 473	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉)
17		mapxpen 8126	. . . . 5 ⊢ ((2𝑜 ∈ On ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐴)))
18	14, 15, 16, 17	mp3an2i 1429	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐴)))
19		entr 8008	. . . 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 693	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐴)))
21	20	ensymd 8007	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2𝑜 ↑𝑚 (𝐵 × 𝐴)) ≈ (𝒫 𝐵 ↑𝑚 𝐴))
22		entr 8008	. 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 693	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑𝑚 𝐴))

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990  Vcvv 3200  𝒫 cpw 4158   class class class wbr 4653   × cxp 5112  Oncon0 5723  (class class class)co 6650  2_𝑜c2o 7554   ↑_𝑚 cmap 7857   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956

This theorem is referenced by:  enrelmapr  38292  enmappw  38293