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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 37320 for a demonstration of an 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 7937 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)) | |
2 | pwen 8018 | . . . 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 6858 | . . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × 𝐴) ∈ V) | |
5 | 4 | ancoms 468 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × 𝐴) ∈ V) |
6 | pw2eng 7951 | . . . 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 7894 | . . 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 691 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐴))) |
10 | pw2eng 7951 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵)) | |
11 | enrefg 7873 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) | |
12 | mapen 8009 | . . . . 5 ⊢ ((𝒫 𝐵 ≈ (2𝑜 ↑𝑚 𝐵) ∧ 𝐴 ≈ 𝐴) → (𝒫 𝐵 ↑𝑚 𝐴) ≈ ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐴)) | |
13 | 10, 11, 12 | syl2anr 494 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑𝑚 𝐴) ≈ ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐴)) |
14 | 2on 7455 | . . . . 5 ⊢ 2𝑜 ∈ On | |
15 | simpr 476 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊) | |
16 | simpl 472 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
17 | mapxpen 8011 | . . . . 5 ⊢ ((2𝑜 ∈ On ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐴))) | |
18 | 14, 15, 16, 17 | mp3an2i 1421 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((2𝑜 ↑𝑚 𝐵) ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐴))) |
19 | entr 7894 | . . . 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 691 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐵 ↑𝑚 𝐴) ≈ (2𝑜 ↑𝑚 (𝐵 × 𝐴))) |
21 | 20 | ensymd 7893 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2𝑜 ↑𝑚 (𝐵 × 𝐴)) ≈ (𝒫 𝐵 ↑𝑚 𝐴)) |
22 | entr 7894 | . 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 691 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ≈ (𝒫 𝐵 ↑𝑚 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 𝒫 cpw 4108 class class class wbr 4583 × cxp 5036 Oncon0 5640 (class class class)co 6549 2𝑜c2o 7441 ↑𝑚 cmap 7744 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-1o 7447 df-2o 7448 df-er 7629 df-map 7746 df-en 7842 |
This theorem is referenced by: enrelmapr 37312 enmappw 37313 |
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