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Mirrors > Home > MPE Home > Th. List > Mathboxes > enrelmap | Structured version Visualization version GIF version |
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 42352 for a demonstration of a natural one-to-one onto mapping. (Contributed by RP, 27-Apr-2021.) |
Ref | Expression |
---|---|
enrelmap | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑m 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpcomeng 9015 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)) | |
2 | pwen 9101 | . . . 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 7689 | . . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × 𝐴) ∈ V) | |
5 | 4 | ancoms 460 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × 𝐴) ∈ V) |
6 | pw2eng 9029 | . . . 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 8953 | . . 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 585 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2o ↑m (𝐵 × 𝐴))) |
10 | pw2eng 9029 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ≈ (2o ↑m 𝐵)) | |
11 | enrefg 8931 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) | |
12 | mapen 9092 | . . . . 5 ⊢ ((𝒫 𝐵 ≈ (2o ↑m 𝐵) ∧ 𝐴 ≈ 𝐴) → (𝒫 𝐵 ↑m 𝐴) ≈ ((2o ↑m 𝐵) ↑m 𝐴)) | |
13 | 10, 11, 12 | syl2anr 598 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑m 𝐴) ≈ ((2o ↑m 𝐵) ↑m 𝐴)) |
14 | 2on 8431 | . . . . 5 ⊢ 2o ∈ On | |
15 | simpr 486 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊) | |
16 | simpl 484 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
17 | mapxpen 9094 | . . . . 5 ⊢ ((2o ∈ On ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((2o ↑m 𝐵) ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴))) | |
18 | 14, 15, 16, 17 | mp3an2i 1467 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2o ↑m 𝐵) ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴))) |
19 | entr 8953 | . . . 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 585 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑m 𝐴) ≈ (2o ↑m (𝐵 × 𝐴))) |
21 | 20 | ensymd 8952 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2o ↑m (𝐵 × 𝐴)) ≈ (𝒫 𝐵 ↑m 𝐴)) |
22 | entr 8953 | . 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 585 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑m 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3448 𝒫 cpw 4565 class class class wbr 5110 × cxp 5636 Oncon0 6322 (class class class)co 7362 2oc2o 8411 ↑m cmap 8772 ≈ cen 8887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-en 8891 |
This theorem is referenced by: enrelmapr 42344 enmappw 42345 |
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