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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 39249 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 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)) |
Colors of variables: wff setvar class |
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 2oc2o 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 |
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